The observable evolution seems to be Latin → French → German → Russian → English

The Russian source is Latin with perhaps a detour via usage in German organsational forms.

The precursor to the Socialist party SPD in Germany has a documented use of this word Präsidium for its organisation going back to at least 1865.

The communist agenda and organisational forms were much stronger and better developed in Germany than in Russia. Lenin had so much a knack for that, and the Prussian/German organisations in general, he described ideal communism as to be run like a German post office:

[…] but were able to decide what to do via a routine comparable, as Lenin famously argued, to the functioning of the German post office.
_{— Christopher Read: "Retrieving the Historical Lenin", in: Ian Thatcher (Ed) "Reinterpreting Revolutionary Russia Essays in Honour of James D. White", PalgraveMacmillan: New York, 2006. (Also compare Lenin: "State and Revolution".}

Russian etymological explanations

Note that Russian Wikipedia has it defined simply as:

Presidium (from Lat. Praesidium — chairmanship) can mean the following:

A group of persons elected to lead the meeting, meeting.
Permanent governing body of state bodies, party, scientific and other public organizations.

And

Etymology Comes from lat. praesidium "protection; cover, protection", then from praesidere "to sit in front, to preside", then from prae- "before"(goes back to the Proto-Indo-European * prai- ) + seēre "to sit , sit"(goes back to the Proto-Indo-European * sed-" to sit ") …

While noting this as a "stable combination"/collocation:

Presidium of the Supreme Council – The very Presidium of the Supreme Soviet

The intermediate step from Latin over German to Russian is then explicated in some Russian etymology dictionaries:

The origin of the word presidium

Presidium . The word is borrowed from the German language, where Praesidium goes back to the Latin praesidium (literally — "sitting in front"; see president ).

_{— The origin of the word presidium in the etymological online dictionary of G.A. Krylov}

Presidium. Latin "presideo" ("sidere" - "to sit") could mean "I preside", and "I protect", and "I rule". From him came our two words - "presidium" and "president".

_{— The origin of the word presidium in the etymological online dictionary of L.V. Uspensky}

Presidium . Borrowing in the XX century. out of it. lang., where Praesidium <lat. praesidium (literally — "sitting in front"). See President.

_{— The origin of the word presidium in the etymological online dictionary of Shansky N.M.}

The last entry is unique insofar that it claims an Italian origin and places the import into the 20th century. Both of these 'facts' seem incorrect.

Other Russian dictionaries emphasise the Latin origin, while it might be noteworthy that it was not the Soviets who introduced the word as late as in 1922:

Presidium Dictionary of Foreign Words
(lat.). Chairmanship of a public assembly, parliament, etc. (Source: "Dictionary of foreign words included in the Russian language". Chudinov A.N., 1910)

Which was the oldest Russian dictionary available to this research.

Apparent earliest use in Russian Socialist circles, late 19th century

Moreover, even the First Congress of the RSDLP in March 1898: 'Documents and materials' uses this word (p222). This may well be still not really the first use of that word, but it is in scanned book and clearly before 'XX century' as one Russian dictionary would have it.

An even earlier use in Ukrainin Osnova Stauripighia, 1872 but that seems unconnected to any socialist aspirations. Further, the scans or too bad too be certain of anything.

For a collection of relevant Russian dictionary entries, see here.

Although a Soviet encyclopedia also again looks onto the Latin source:

Modern explanatory dictionary ed. "Great Soviet Encyclopedia"

PRESIDIUM

(from Lat. praesidium, lit. - protection; re. - chairmanship), a group of persons elected for the collegial conduct of a meeting, meeting; the governing body of some political, social and other organizations.

From Krylov's etymological dictionary of the most common Russian words:

The until now most complete Russian printed Russian etymological dictionary by Max Vasmer does not list presidium itself, but notes that president was also imported from German:

Президент 'Präsident, Vorsitzender’, seit Peter d. Gr., s. Smirnov 236. Über nhd. Präsident aus lat. praesidens. Daneben wurde gebraucht: презус 'Vorsitzender eines Kriegsgerichtes’ (D.), älter презес 'Vorsitzender’ Peter d. Gr., s. Smirnov 236. Über nhd. Präses (18. Jhdt.) oder poln. prezes aus lat. praeses G. praesidis, s. Schulz-Basler 2, 646.
_{(Vol II, p242)}

German use since 16th century

The etymology for the German Präsidium is listed as:

preside Vb. to 'preside over, lead (an assembly)', borrowed (16th century) from the Latin praesidēre '(to) protect, cover, preside over, be in the lead, lead', actually 'preside over, preside over', or from the old French presider based on this; cf. the Latin sedēre, 'sit, hold a meeting'. President m. 'chairman or head of an assembly, important state or social organs, institutions, head of state of a republic' (16th century), […]

In workers associations the use of Präsidium for collective leadership committe must go back to at least 1848, like in the Cologne newspaper: Zeitung des Arbeiter-Vereines zu Köln, Volume 4, 9th May 1848. (gBooks).

Latin origin

So we need to go to just the Latin root for it, which gives us the exact same root as used for the American President:

praesidēns (genitive praesidentis); third-declension one-termination participle

sitting before or in front of
sitting beforehand
guarding, watching, protecting, defending
presiding over, directing, commanding, controlling, governing, superintending (substantive) a president, director, ruler, governor, leader

We see collective chairmen in a leading committee. The use of the spelling presidium is then the probably only Russian thing in this. If English would have adopted the attested earlier German Präsidium, then it is just the spelling getting more complicated. Although presidium and praesidium are both recorded.

English re-import from Russian

Presidium in this context is an English language invention precisely to describe the Soviet communist structure.

As the Oxford English Dictionary records int 2nd edition:

Presidium
The presiding body or standing committee in a Communistic organization, esp. in the Supreme Soviet. Also attrib.

1924: Observer 23 Mar. 13/5 “In a second decree the Presidium of the Union C.E.C. decided to replace the sentence of ten years strict isolation passed on the Catholic Archbishop Ciepliak by the All-Russian C.E.C. by expulsion from the territories of the Union of Socialist and Soviet Republics.”

1927: Glasgow Herald 10 Oct. 11 “Mr Arthur Horner (South Wales), a member of the National Executive of the Miners' Federation, presided, and was supported by a presidium of 11.”

Notice the 1920s English usage outside the Soviet Union for a collective steering committee in Wales to express some kind of delegate equality in strict hierarchy. This word is also used in present day North Korea or China, and correctly noted also in Bosnia-Herzegovina, as the Presidency of Bosnia and Herzegovina is really a team of three chairpersons.

But there seem to be quite a few antedatings for presidium in English as well.

1919 United States Congressional Serial Set, Volume 7599, Bolshevik Propaganda (gbooks)

John Reed: "Ten Days that Shook the World", International Publishers, 1919. (gBooks)

Daily Extracts from the Foreign Press, Volume 25, Great Britain. War Office. General Staff, H.M. Stationery Office, 1918. (gBooks)

Calcutta Review 99, (about German law), 1894. (gBook)

Excursus

The word itself isn't special at all, only a minor spelling variation which acquired only in English the Soviet undertone for one narrowed sense. Therefore, a strictly 'Germanic origin' is only possible via a midway stop towards Russian. It is also found in French, Italian and Mexican Spanish:

presidium, with the meaning of 'table of honour' or 'presidency', is totally incorporated into current Mexican Spanish, especially in the more or less formal language registers, spoken or written. It is a word that does not appear in the general dictionaries of the Spanish language, which certainly include the word presidio, from the Latin praesidium, and which has several meanings: 'garrison of soldiers', 'city or fortress that could be guarded by soldiers', 'penitentiary establishment', 'group of prisoners', 'sentence consisting of the deprivation of liberty', 'aid' or 'assistance'. Obviously none of these meanings has any relation to the meaning of presidium in Mexico, in the expressions which speakers usually use when they begin their speeches: "Mr. Secretary, gentlemen of the presidium, ladies and gentlemen…".

Summary

Most Russian dictionaries prefer in their shortness the origin of presidium as Latin, while a few highlight that German was an intermediate step. German however presents the same phenomenon, emphasising the Latin roots with French being the transmitting language.

German socialist circles used that word from at least the mid 19th century for their collective leadership organisation, and Russian socialists used the russified version of it in their own organisation from at least the late 19 century.

The English re-import of that spelling variation is obviously much older than parallel to 1922 or 1924 as listed by the OED. The chairmen collective leadership sense is older than any post revolution Soviet body.

The chain of origin for this word and its usage with its sometimes slight variations is therefore:

Latin → French → German → Russian → English