This really depends on the source in question. The Anglo-Saxon Chronicle habitually refers to them as The Danes, but also sometimes refers to them as The Enemy or, particularly in earlier entries, The Pagans. In the hagiographical 'Vita Ælfredi written by Asser of Sherborne, the Danes are continially referred to as The Heathen, although of course this is a useful rhetorical device to frame the Danes as perfect villains to Alfred's virtuous Christian hero. In the Welsh chronicle the Annales Cambriae, the Danes are referred to as "'The Northmen" in an entry for 895; distinct from the Northumbrian English, although they are also referred to as Pagans elsewhere in the Chronicle.

In the 10th Century Exeter Book, one poem recalls how its hero went to fight "the wycingkin, which suggests yhat that vernacularly, the English may have already been referring to the Danes as 'Vikings'.

Can only speak for the Irish sources, but we find a range of terms used and these vary a bit over time.

Geinti and gentiles were the most frequently used in the earliest period. This derives from the Latin root of gens (plural gentes) meaning ‘race(s) of people’, with gentilis being a derivative. In monastic circles both were used in the Jewish sense of ‘gentile’ to mean “heathen(s)”.

For instance the first Viking raids in the North of Britain are described in the Annals of Ulster for 794 as:

Uastatio omnium insolarum Britannie a gentilbus (“Devastation of all the islands of Britain by heathens.”)

A year later, in 795 we find:

Loscadh Rechrainne o geinntib (the burning of Rathlin by the heathens”)

The Annals of Innisfallen briefly refer to Geinte i nHerind ("the heathens in Ireland") in 796. From this period onwards both geinte and gentiles are used to describe the Norse Heathens. The Annals of Ulster uses the term numerous times over the next century and a half. In 942 we find do ghentibh Atha Cliath ("by the heathens of Dublin"). With a few other late examples too.

The other main term is simply “The Foreigners”, or Gaill. This was obviously a general term in Irish and it would be later be used to refer to the English too, but it’s application to the Scandivanians first occurs in 828 with:

“A great slaughter of porpoises on the coast of Ard Cianachta by the foreigners (ie. ‘o Gallaibh’)

Similarly we begin to find particular groups of vikings connected to the towns they founded, e.g. Gaill Atha Cliath (“the foreigners of Dublin) who are first noticed by the Annals of Ulster in 845. In 916-18 we find Gaill Puirt Lairge (“the foreigners of Waterford), and so on. It took time for gaill to overtake geinti as the term of choice.

We also find an awareness of competing groups amongst these viking foreigners. We find reference to Dubgaill (‘black/dark foreigners) and Finngaill (‘white/fair foreigners). The Annuals of Ulster notes in 851 that the dubgennti (‘dark heathens’) drove the finngail out of Dublin. There’s some debate about this in historiography. Some take the view that the Dubgaill were Danes and Finngaill Norwegians. Others think of them as more specific factions, followers and descendents of particular warlords. Either way there are references to two distinct groups of 'vikings', as well as distinctions between the ‘foreigners’ of different towns.

We also find other terms, albeit not as common. Since you wonder about “Northmen”, we do find some reference to Nordmenn (Norwegians) and Danir (Danes). Danir doesn’t appear until the late tenth century, but we find Nordmenn in the mid-ninth. In 837 the the Annals of Ulster note that:

Longas tre-fhichet long di Norddmannaibh for Boinn… (“A naval force of the Norsemen sixty ships strong was on the Bóinn…”).

This is the only example in Irish, but we also find a handful of Latin inflections such as Nordmannorum. We find Old Norse Danir borrowed into Irish as Danair in a few brief instances in the late tenth century (just five times between 986 and 990). In 986 the Annals of Ulster records:

Na Danair do thuidecht i n-airer Dail Riatai, i teora longa coro riagtha secht fichit diibh & coro renta olchena (“The Danes arrived on the coast of Dál Riata, that is, with three ships, and seven score of them were executed and others sold.)

Í Coluim Cille do arcain do Danaraibh aidhchi Notlaic coro marbsat in apaidh & .xu. uiros do sruithibh na cille (“Iona was plundered by Danes on Christmas Eve, and they killed the abbot and fifteen men of the seniors of the church”)

However it’s not entirely clear from the annals why this term would show up in the span of a few years and disappear again. The most common terms were undoubtedly ‘foreigners’ and ‘heathens’, until many centuries after the fact.

There's been a lot of good conversation about the terms our sources use, and how mixed the terms used actually are, but I want to tackle a slightly different angle of approach - namely, the assumption that raids were coming in from the east, not the North. This is definitely sometimes true, but not as universally true as you might think.

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Sometime in the late 800s, saga evidence says that a man named Ketill Flatnose left from Norway and settled in the Orkney Islands, a place well-known to Norwegians from earlier raids. His daughter would eventually convert to Christianity before moving first to the Faroes and then to Iceland, becoming one of the leading settlers of that country. But, Norwegian presence in the Orkneyjar was well-established. According to Orkneyinga saga, a fairly early but extremely difficult to work with text, there was a kingdom in the Isles generations before Ketill, conquering much of northern Scotland and harrying the coasts seasonally. This would continue for a very long time - According to Njals saga (among other sources), one of the Viking armies that fought at the Battle of Clontarf (1014, in which Brian Boru of Ireland was killed) sailed from Orkney. Even into the 12th century, Orkney would remain part of Norway, albeit a fairly estranged backwater - Harald gilli, a contestant in the Norwegian civil wars of the 12th century, traveled from the Isles, claiming to have been born on the expedition that led to Magnus Barefoot's death in 1103 (which traveled to Ireland through Orkney). Even later, in 1263, Hakon IV Hakonarson traveled to Orkney as a power base to fight against Scotland - another case of a Norwegian king dying on a trip to the isles......

Anyway, this strongly suggests that a (not the only, but one) common route to travel from western Scandinavia to England was by sailing directly across from the Vestfold to the Shetlands, then down to the Orkneys, and then from there raiding England. (Jan van Nahl's work on navigation in the Viking Age suggests that holding a single latitude across open ocean was very manageable in the Viking Age, and probably more reliable than island hopping in the far north Atlantic).

And, well... boats sailing from Orkney for a season of raiding come in from the north, not the east.