Where in the sources does it say that Syagrius claimed to be "merely governing a Roman province"?

@MAGolding is mostly right in his answer -- especially the part that we don't know what title Syagrius used for himself -- but we can be fairly confident there was one he did not use -- emperor.

Consider the background of his rump state. After the death of Valentinian III, & the chaos that followed, the general Ricimer took control of Rome & her garrison, & became the man who picked emperors. His father Aegidius, in command of an army in Gaul, was one of several who refused to recognize Ricimer or his puppet emperors, & revolted against him. Now this situation was largely so much posing, because neither Ricimer nor Aegidius were in a position to do much about the other (IIRC the chronicler Hydatius does mention that Aegidius attempted to march on Ricimer, but was unable to proceed past the boundaries of Gaul), however Aegidius was faced by the threat of Germanic barbarians inside the nominal boundaries of the Empire. These included the Visigoths & Burgundians. They claimed to be loyal supporters of the true Roman emperor, & if he claimed the purple they could use this as an excuse to attack him.

Aegidius' best move would be to acknowledge the remaining emperor, the one in Constantinople, as his master. This would not only give him legitimacy, and would prompt the grant of a title from that emperor giving him some measure of prestige his rivals lacked, it now became politically inconvenient for these barbarians to attack him. While the emperor was too far away in Constantinople to offer him any real help, he did take some interest in the affairs of Gaul & Spain; IIRC, the letters of Avitus of Vienne, who handled the foreign correspondence of the king of the Burgundians in the late 5th century, includes several letters written to the eastern emperor or his representative, who dangled prestigious although empty titles before the Burgund king in return for favors.

If his father did embrace this solution -- give lip service to the emperor in Constantinople -- Syagrius would continue this: acknowledge the emperor in Constantinople as his liege in return for a title. This worked well enough until Clovis became king of the Franks. Clovis was more interested in ruling Gaul than being a peaceable subject of the Roman empire. After consolidating his position amongst the Franks, he was able to not only destroy Syagrius but overthrow his chief rival in the Battle of Vouillé (Alaric II king of the Visigoths), who was nominally loyal to the eastern emperor. With no recourse to Clovis' actions, the emperor could only acquiesce to the new situation, & made Clovis an honorary consul and patrician.

In short, Syagrius could not have ruled his share of Gaul had his father or he used the title "emperor", because that would both destroy the prestige he received from the connection, & make him vulnerable to destruction as a rebel.

To make a long answer short:

Nobody knows what title(s) Syagrius actually used.

Part One (of Ten): Imperial titles.

I note that Greek texts tended to refer to the Roman Emperor as Basileus, which originally meant rex or king, but later tended to mean emperor, from early in the empire.

I have also read that some late Roman writers tended to refer to Roman Emperors as reges "kings". I note that Emperor Constantine I appointed his nephew Hannibalianus Rex Regum et Ponticarum Gentium, "King of the Kings and of the Pontic People".1 in 337.

https://en.wikipedia.org/wiki/Hannibalianus 2

If the Romans got over their horror of Hannibal by then they might have got over their hate for the title of king.

Part Two: Royal Titles:

I also note that though Odoacer and later Theodoric and his successors used the title of rex, "king", there were three different ways the title of rex could be used in Medieval times.

A king might use the simple and plain title of rex, meaning "King" or "the king", apparently assuming that everyone would know which kingdom he ruled.

Or a king might use the title of king of an ethic group.

Or a king might use the title of king of a land area.

So I ask whether Odoacer or Theodoric ever used the title of "King of Italy" or instead used the title of king as leader of their Germanic tribesmen, an ethnic kingship, while claiming territorial rule over Romans in Italy as Roman officials with Roman titles.

My answer here: https://history.stackexchange.com/questions/47888/why-did-odoacer-choose-to-be-king-instead-of-emperor[1] also asks that question.

Part three: Syagrius as a Possible Roman officer:

In any case a Roman ruler of a detached region like Syagrius in northern Gaul could either claim to be a Roman official loyal to an emperor, or else claim to be an emperor himself.

So Syagrius might have claimed to be the successor of his father Aegideus as magister militum per Gallias or assumed some other title, or maybe even been granted such a title by a Roman Emperor or a Roman usurper.

Part Four: A List of Changes of Emperors During Six Important Years From 474 to 480.

In 474 the Eastern Emperor Leo I sent his relative Julius Nepos to depose the western Emperor Glycerius who he considered an illegitimate ruler. Leo I died in 474 and was succeeded by his grandson Leo II and his son-in-law Zeno.

In 475 western emperor Julius Nepos was deposed by Orestes, who made his young son Romulus Augustulus emperor in the west. Julius Nepos fled to Dalmatia. Basilicus deposed Zeno and made himself eastern emperor in 475 to 476.

In 476 the Germanic soldiers under Odoacer deposed Romulus Augustulus and the Roman senate sent a message to Constantinople saying that one emperor would be enough for the entire empire from now on. Zeno continued to recognize Julius Nepos as the rightful western Emperor until Nepos was assassinated in 480, perhaps at the instigation of Odoacer and/or Glycerius.

Part Five: Later Imperial Claimants in the West.

There were several claimants of the western imperial throne after 476/480.

Burdunellus became a "tryant" in Spain in 496 (but was soon defeated and killed), which should mean that he claimed to be emperor. Peter became a "tryant" in Spain in 506 (but was soon defeated and killed), which should mean that he claimed to be emperor.

https://en.wikipedia.org/wiki/Burdunellus 3

https://en.wikipedia.org/wiki/Peter_(usurper) 4

Part Six: "Emperor" Masties.

When the Vandals took over Roman North Africa in the early 5th century, a number of small Romano-Berber states emerged where the Vandals didn't manage to take over. Some of those states endured until the Muslim conquest in about 650 to 700.

A man named Masties was a ruler of the Kingdom of Aures in eastern Algeria and part of Tunesia. Mastises supposedly ruled for 67 years from c. 426-494 or from 449-516.

According to an inscription found at Arris, Masties reigned for 67 years as a dux, and 40 years (or only 10 years) of them as an Emperor of "Romans and Moors" until 516 AD, where he knew how to practice a skillful policy to balance between the Byzantines and the Moors.4 There is no indication that the "empire" of Masties was recognized by Constantinople, in which the Berber princes were considered as "usurpers".

https://en.wikipedia.org/wiki/Masties 5

If Masties used the title of Emperor for 40 years, he would have started using it between 454 and 476, and if Masties claimed to be emperor for ten years, he would have started using the title between 484 and 506.

Part Seven: List of Dates When Masties Might Have Claimed the Imperial Title.

Valentinian III was killed in 455, the last emperor descended from the Theodosian dynasty.

Petronius Maximus was killed in 455.

Avitus was deposed in 456.

Marcian (east) died in 457, the last emperor connected by marriage with the Theodosian dynasty.

Majorian was killed in 461.

Libius Severus died in 465.

Anthemius was killed in 472.

Olybrius died in 472.

Glycerius was deposed in 474.

Leo I (east) and Leo II (east) died in 474.

Julius Nepos was deposed in 475 but continued to reign in Dalmatia until 480.

Zeno (east) was deposed by Basilicus in 475.

Romulus Augustulus was deposed in 476.

Basicilus (east) was deposed by Zeno in 476.

Julius Nepos was assassinated in 480.

Syagrius in Gaul might have been deposed and killed in 486.

Syagrius in Gaul might have been deposed and killed in 487.

Zeno (east) died in 491.

Syagrius in Gaul might have been deposed and killed in 493.

Syagrius in Gaul might have been deposed and killed in 494.

Usurper Burdunellus in spain was killed in 496.

Usurper Peter in Spain was killed in 506.

So Masties might have used the extinction of the Theodosian Dynasty, or the overthrow of an emperor by a usurper or a barbarian, to justify claiming the imperial throne, sometime during the period of 454 to 506.

Or possibly another Romano-Berber ruler used the Imperial title in Africa without any evidence surviving to the present, and Masties took the title when that other "emperor" died.

Part Eight: Emperors in Britain?

The western Roman Empire lost control of Britain during the usurpation of Constantine III in 407-411, and never regained it except possibly for unproved brief occupation of parts. More than a century later Procopius wrote that since that time Britain had been ruled by "tyrants", which should mean Roman usurpers not recognized by the Emperors in the west or the east.

So it is possible that Syagrius might have recognized the authority of emperors in Britain, instead of or in addition to the western and/or eastern emperors. Or maybe Syagrius changed his allegiance one or more times.

Part Nine: Syagrius as Emperor or Usurper?

Or Syagrius might have claimed to be emperor himself during part or all of his rule.

Historians have mistrusted the title "rex Romanorum" that Gregory of Tours gave him, at least as early as Godefroid Kurth, who dismissed it as a gross error in 1893. The common consensus has been to follow Kurth, based on the historical truism that Romans hated kingship from the days of the expulsion of Tarquin the Proud; for example, Syagrius' article in the Prosopography of the Later Roman Empire omits this title, preferring to refer to him as a "Roman ruler (in North Gaul)". However, S. Fanning has assembled a number of examples of rex being used in a neutral, if not favorable, context, and argues that "the phrase Romanorum rex is not peculiar to Gregory of Tours or to Frankish sources", and that Gregory's usage may indeed show "that they were, or were seen to be, claiming to be Roman emperors."5

https://en.wikipedia.org/wiki/Syagrius 6

S. Fanning, "Emperors and empires in fifth-century Gaul", in John Drinkwater and Hugh Elton, Fifth-Century Gaul: A Crisis of Identity? (Cambridge: University Press, 1992), pp. 288–297

Part Ten: Conclusion.

So this long answer shows that nobody knows what titles Syagrius used during his rule, but there are interesting possibilities.