What exactly did the 1868 Szepes Petition do?

There is indeed a nice diploma thesis about this, that @WiJama found:

Lucia Tatarková; "Vplyv Spišských petícií na národné hnutie Slovákov v 60. rokoch 19. storočia", Filozofická fakulta Univerzity sv. Cyrila a Metoda v Trnave, 2007. (doc) / (The Impact of Spiš Petitions on the National Movement of Slovaks in the 60s of the 19th century)

It has a German abstract attached and that translated into English reads:

The aim of this work is to present the political development in Hungary in the sixties of the 19th century, as well as its impact on the life and national life of the Slovak population.

The detailed analysis offers a picture of the difficult conditions in which the Slovaks found themselves during their common life with Hungary. The inhabitants of the Spiš region tried to improve the difficult situation under the Hungarian oppression. This work is dedicated to the analysis of three Zipser petitions written on the basis of these facts. With the help of archival materials and the press of the time, I tried to point out the general activities of the Spiš people during the time that had not been researched too much.

The Spiš petitions did not influence the actual situation of the Slovak population in Hungary, but they clearly aroused the interest of the population for cooperation in the Slovak national movement.

It concludes with

During these ten years, several proposals for improving the position of the Slovak nation were created and submitted, but neither was accepted. The nationality law issued at the end of 1868 stopped any action aimed at gaining the Slovak national rights. A direct path to violent Hungarianization has opened, which has steadily increased over the next few decades.
One of the most important documents requesting Slovak national rights is the inclusion of three Spiš petitions. Despite the fact that petitions from 1863 and 1864 contained requirements concerning only the Spiš territory, they were indeed a unique step in the Slovak population. The last Spiš petition from 1868 was a real national petition comparable to the Memorandum of the Slovak Nation. Very precisely formulated requests related to the whole nation and, if they were fulfilled, the lives of Slovak citizens in Hungary could actually develop in a different direction.
Slovak historians themselves present only two documents in their works. The 1864 Spiš Petition escaped their attention despite the fact that it was no less important.

However, three attempts by the Spiš Slovaks could not influence or improve the position of the Slovak nation to a greater extent. Despite the well-conceived requirements contained in these documents, their expected fulfillment has not been achieved. Like other national applications, they were only formally equipped, and no one was involved in them. As a result of their writing and submission, a new wave of allegations of anti-possession activities was raised. The observation and suspicion of panslavist agitations were not avoided by many representatives of the Spiš region, including the significant figure of Slovak history, Martin Čulen. The then director of the Royal Catholic Grammar School in Levoča took only insults and accusations from his post. As the main accused of writing the third Spiš petition and spreading panslavist ideas, he found himself in an early retirement without proper interrogation, whereby the Spiš Hungarian government reached a halt to the further spread of the national movement in this area. However, even by examining archival materials, it is not possible to confirm with certainty that the director of Levoča really took part in this movement.

The Spiš petitions nevertheless had one undeniable significance. So far, little nationally awakened Spiš has joined this nation in this step. These documents clearly show that the inhabitants of the Spiš region are becoming aware of their belonging to the Slovak nation. They became convinced Slovaks, as evidenced by their overall involvement in national nationwide events and their support. It is therefore evident that the national movement in the peak period of the 1860s hit the entire territory of Hungary inhabited by the Slovak nation from west to east.

The actual text of the 1868 petition contains quite a lot of 'reasoning' (or fluff that draws on nationalism) and then comes down to very precise demands. For example, on national(-ist) recognition:

1. The Slovak nation will be recognized in Hungary as equal to the Hungarian one recognized and recognized by law.
2. Seats or county shall be rounded by nationality.
3. Every throne, every speech, shall be in it, and in this all negotiations and negotiations shall be done. The minutes of the committee and general meetings of the stools are to be held in the speech of the people of the stool, which are then to be placed at the appropriate higher places from the seat of the nobility and with the attached passage to the Hungarian language.
4. The ordinary courts are also supposed to use the Slovak language, not only in investigations and statements, but also when speaking and negotiating in Slovak stools.
5. In Slovak seats, only such officials have to establish who, among other requirements, have full knowledge of Slovak language, and are perfectly versed in this.
6. At the highest provincial nobility, according to the relative numbers, the Slovak men should be established, but one Slovak section, which would represent the interests of the Slovak nation, should be in each of the ministries.
7. The Slovak nation should have the right to use the red and white national flag and its national white-blue-red flag.

Or on administration:

I. About the Seats

1. In the sear committees, each municipality should elect its representatives according to the population. Smaller municipalities in conjunction with the election.
2. Those corporations that have their own and special administration and chairmanship do not pertain to the right of chairmen to vote, and to blend in the chair.
3. Civil servants, under the responsibility of the Committee, shall not have the right to vote and to take decisions. Honorary Chancellors should stopp.
4. When electing deputies, a commission appointed by the committee from one municipality to another, and not just descriptions of the elected voters, is to be appointed by the chairman, to collect the lie and the votes of the deputy. To the commission, the candidate chooses two confidants who are rewarded to be members of the public treasury commission.

The second point contains four requirements concerning the modification of the district administration. The composers were primarily concerned with ensuring the public and political position of Slovaks in the seat offices. Above all, the new order for the election of county deputies should have prevented the various inconveniences that often occurred.

_{(Relying for a large part quite heavily on machine translations. Prepare your salt shaker. Or if speaking Slovak, please edit)}

Background:

Thus, in Hungary, alongside the natio hungarica, embrac- ing persons of Slovak, Serb, Ruthenian, and Rumanian speech as well as the Magyar-speaking majority, the natio croatica was recognized because Croatia possessed a separate administration and its own historical nobility. (The Croats used Latin as their official language as did the Hungarians.) In existing conditions there could be no Slovak nation: those Slovak speakers who enjoyed noble rank belonged to the natio hungarica, while the rest, according to official theory, be- longed to the plebs along with all other non-noble inhabitants of the kingdom, including those whose mother tongue was Magyar. […]

In Hungary, where traditionally only nobles had enjoyed political rights in the state and only state-nations were endowed with the attributes of nationhood, the Slovak national movement had keenly felt the lack of a truly Slovak noble class. The time now appeared ripe for a rapprochement. The Slovak nationalists were once again turning their gaze back from Vienna to the Hungarian scene, while the Slovak nobility seemed to be finding its way back from 'Hungarianism' to its Slovak roots. But the apparently favourable circumstances proved deceptive. The older generation of the Slovak-speaking nobility still clung desperately to their antique Hungarian patriotism while the younger nobles were fast losing the last remnants of Slovak sentiment and assimilating to Magyardom.
_{Peter Brock: "The Slovak National Awakening: an essay in the intellectual history of east central Europe", University of Toronto Press: Toronto, Buffalo, 1976.}

During the era of Compromise (1867), the 'Slovak question' had disappeared completely from the official deliberations, and finally the Nationality Law drafted by Ferenc Deák and Baron Josef Eötvös declared only a "unified and indivisible Magyar nation" as a state people. In the eighties, the fateful Magyarization policy developed on this newly created legal basis, which the Slovaks opposed with the Spiš petition and the founding of Matica Slovenská.

The Nationalities Act (G.A. 44 of 1868) only refers to the 'confession of a nationality' as a personal characteristic. It included rights of association and language rights, and so it was left to the administration to interpret the extent to which Slovak associations were suspected of being hostile to the state or not.

'Great Slav' or pan-slavic sympathies and russophilia were regarded as the main accusations which led poets such as Viliam Pauliny-Tóth to go into semi-legality. Since Béla Grünwald's agitation, the Magyarization policy came in full force against the Slovak movement and inspired Kálmán Tisza's government with the elitist cultural and power consciousness of the 'Hungarian race'.
_{(Src Ludwig von Gogolák: "Beiträge zur Geschichte des slowakischen Volkes III. Zwischen den Revolutionen (1848—1919)", (Buchreihe der Südostdeutschen Historischen Kommission 26), Oldenbourg, München: 1972.)}

It seems very noteworthy to mention that the region that gave the petitions in question their name are within Slovakia now, and had Slovaks, were in Hungary then, and had Hungarians, but were in cities quite heavily settled with ethnic Germans; and more important: linguistically German dominated (Zipser Sachsen).
_{(Cf. Pavel Kulha: "Polyethnizität und sozialer Status: Stadtbürgertum in Pressburg und Leutschau in der zweiten Hälfte des 19. Jahrhunderts", Dissertation, Leipzig, 2016. (PDF, p24–55.))}

Additionally, significant Czech and Polish speaking people would add to the multi-ethnic mix of settlement in this border region.

This thesis
_{(Marcel Jesenský: "Between Realpolitik and Idealism: The Slovak - Polish Border, 1918-1947" University of Ottawa, 2012, PDF)}
sheds additional light on attempts by the Slovaks to obtain the same standing in the Empire as the Poles. Note that Szepes was spelt Scepusium through much of the 19th century - which complicated Google searches.
_{– Pieter Geerkens}

The later confrontational stance from both sides on nationalism may sound 'bad', but it has to be seen as occurring in a dynamic between these many petitions. With those of Szepes/Zips/Spiš to be read in context:

Even after the optimistic atmosphere of the March days gave way to paranoia and con ict, the Slovak leadership retained its Hungarian loyalties. In November 1848, the Slovak National Council’s “Challenge to the Slovak Nation” proclaimed that the Slovak volunteers were ghting for “the unity of the Hungarian monarchy,” though this Hungarian monarchy was to exist in a Habsburg context: the Magyars were accused of trying “to detach our land from Austria, so they can rule over us by themselves … let the king rule over us with law and justice.”66 Only on 19 March 1849 did Slovak patriots finally demand separation from Hungary: 28 Slovak leaders went to Olomouc and presented the Habsburg emperor with a petition demanding

1. at the Slovak nation, around 3 million strong, be recognized as a nation within well-de ned state borders. We demand that country, which we have inhabited since the most ancient times, where our own language is normally used domestically and in public life, be united from many countries into a united political whole which has never ceased to be called a Slovak country, Slovakia.

The petition then proclaimed equality between the various nations in the Austrian Empire, and demanded the end of Hungarian-language administration and a Slovak constitution. Yet even this petition stressed the “most ardent wish to see the Austrian monarchy uni ed and powerful.”68 When Kossuth’s extremism forced Slovaks to choose between linguistic loyalties and Hungarian political loyalties, several Slovak patriots chose to break with Hungary. Nevertheless, they were only willing to make this choice when forced by extreme circumstances. Most Slovaks, through most of the Revolution, sought to secure linguistic rights within Hungary. Even the March 1849 petition shows no desire for Slovak statehood: the idea of an independent Slovak political unit apparently had not occurred to anyone. Instead, Hungaro-Slavism was brie y transformed into Austro-Slavism. A er the revolutionary confrontation abated, however, Slovak nationalists returned to their traditional Hungaro-Slavism. One key gure in the post- Revolutionary period was Daniel Lichard, a proli c journalist with Vienna’s Slovenské noviny [Slovak Newspaper], and author of literally hundreds of titles. In an 1865 pamphlet, Lichard has a ctional parson explain the necessity of supporting the Matica slovenská to various skeptics and doubters. Lichard makes clear that Slovaks were also Hungarians [uhri].

[A]ll the inhabitants of this country: we who are called Slovaks, additionally our brothers the Serbs, Croats, Russians; Magyars, Germans, Romanians, who are commonly called Olasi if they live in Hungary — we are all Hungarians, because we live in the Hungarian country and live under the same laws.

The fictional parson then contrasts “the political word Uhor and the national or linguistic concept of maďar. In this political sense, we Slovaks too are Uhri; but from the national point of view we cannot be and are nothing else but Slovaks.”
Lichard was not the only intellectual attempting to teach Hungaro-Slavic concepts to Hungary’s Slovak peasantry. Ondrej Radlinský’s school primer, the 1871 Školník, described the Slovak homeland as follows:

We Slovaks have in which state and country our homeland? In the state of Hungary-Austria, and particularly in the country of Hungary.

Notice that while neither half of this two-tiered formula bears a Slavic ethnonym, both halves mention Hungary. Slovaks patriots saw no distinction between Slovak and Hungarian loyalties: the one implied the other. is is most clearly shown in the “Newest Slovak Song,” published in an 1871 Slovak Almanac:

Uhorsko jest krajna krásna Hrom w ňej ljeta z čísta, jasna Bo w ňej jesú dwe ďeťičky, Krásne, čo hore hwieyďičky ... W tam Uhorsku jest Slovensko Slowákow wlastnje wlasťensko: Tam Maďar ňech ňepanuje. Nech sa swojim ukojuje

Hungary is a beautiful land
with thunder clean and clear
There are two children in her beautiful,
like stars above …
In this Hungary lies Slovakia
The Slovaks’ own homeland:
There let the Magyar not rule
Let him take care of his own affairs

_{Alexander Maxwell: "Choosing Slovakia. Slavic Hungary, the Czechoslovak Language and Accidental Nationalism", I.B.Tauris: London, New York, 2009.}