I think this is a difficult but good question to answer.

It is true that the Norsemen (we don't unfortunately have many evidences by ON vikingr, the plunderer in a narrower sense) certainly called themselves as Norð-menn (lit. trans. in English: the Northmen) during the Viking Age, but the primary meaning of this ON word seemed to focus rather on the inhabitants of now Norway (i.e. the now Norwegians), not the Scandinavians or the raiders from Scandinavia in general, as Latin Nortmanni usually connoted. In short, the concept of ON Norðmenn did not correspond exactly with Latin Nortmanni in continental sources.

Let us see some examples to explore this difference further:

 

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1: The Account of Ohthere the Northman from Hålogaland, Northern-Norway (ca. 900)

Old English translation of Orosius, compiled in the court of Wessex in the beginning of the 10th century, includes some additional accounts of the Northern Europe, collected from visitors of the court of King Alfred the Great around ca. 900. Among them, we find a 'Northman' called Ohthere (ON: Ottar) came from Northern Norway, though he apparently did not engaged in the raiding activity by himself. So, how he describe his origin as well as the circumstances of diverse peoples in Scandinavia?

• 'Ohthere said to his lord, King Alfred, that he lived in the northernmost of all the Northmen (earla Norðmonna norþmost bude)' [Bately & Englert (eds.) 2007: 44].
• He also distinguishes the Northmen from hunter-gathering Finns (Finnas: usually identified with the Sámi people) and the hostile Kvens (Cwenas), and further, mentions the Danes (Dene) and the Svears (Sveon).
• While we don't know what exact criteria (like polity, language, or custom?) Ohthere or the scribe who put his account on the parchment employed to distinguish these peoples, but scholars generally agree that the Northmen in Ohthere's account did not include the Danes and the Svears.

 

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2: Other Contemporary Sources

We don't have enough amount of the extant contemporary sources from Viking Age Scandinavia, so I'll concentrate on the skaldic verse corpus, instead of runic inscriptions below.

To give an example, The poet Eyvindr skáldaspillir Finnsson employs Norð-menn as somewhat broader, integrating concept of various local peoples in now Norway against the 'Island-Danes', in St. 3 of his Hákonarmál (ca. 961): '{The sole slayer of jarls} [= Hákon] called on the Háleygir [the inhabitants of Hålogaland] just as on the Hólmrygir [the inhabitant of Rogaland, SW coastal Norway]; he went into battle. {The munificent terrifier of Island-Danes} [= Hákon] had the good support of the Norwegians; he stood under a helmet of metal.' (English Rough translation is taken from the link, with some additional annotations by me).

I also cite the poem by Steinn Herdísarson's Óláfsdrápa, St. 14 (ca. 1070), on the generosity of King Olaf Kyrre (r. 1067-93) St. Olaf of Norway to Norðmenn, i.e. his subject, the Norwegians: 'The hoard-generous host-leader, who reddens spear-points, gladdens his benchmates with rings; the monarch lets his men rejoice in the gifts. {The enterprising king of Norway} [= Óláfr] gives lavishly to the Norwegians; {the diminisher of the English} [= Óláfr] is generous. Óláfr born [beneath] the sun…'.

The concordance lists 17 cases of such uses of Norðmenn by the poets from the 10th to the 13th century, and all of these cases fit rather well with the translation as 'the Norwegians' than the Scandinavians in general.

As for the collective term for the whole Scandinavians in Old Norse texts, the phrase, 'á danska tungu [those who speak in the Danish tongue]’ might be the closest concept OP mentioned above.

 

Works mentioned:

• Bately, Janet & Anton Englert (eds.). Ohthere's Voyages: A Late 9th-Century Accounts of Voyages along the coasts of Norway and Denmark and its Cultural Context. Roskilde: The Viking Ship Museum, 2007.

[Edited]: corrects the wrong identification of Olaf in Steinn's poem from St. Olaf to Olaf Kyrre.