The 793 raid on Lindisfarne is the first true raid that we have a record of, i.e. that someone wrote down and dated. It's known from two sources: (1) a pair of contemporary letters by the Englishman Alcuin who was then living in France but who wanted to console his monastic brethren after the attack, and (2) an account in the Anglo-Saxon Chronicle, which was initially written about 100 years after these events to show how important the king of Wessex, Alfred ('the Great'), was for defeating these Norse-speaking invaders; someone in the north of England got a copy of this Chronicle and added in a few lines about Lindisfarne to show that the north had also suffered and played a central role in the conflicts with these raiding parties and later invaders.

So Lindisfarne is remembered as the first attack because we've got a few contemporary letters and a much later chronicle account, all of which give a vivid picture that's fun to study. The original edition of the Anglo-Saxon Chronicle was written in the 890s and it actually dates the beginning of problems to 787, when a trio of ships lands, the king's reeve (sort of a sheriff/customs control) goes out to bid them to go check in at the king's town, and they kill him instead. A later chronicler would clarify that this event occurred at Portsmouth on the English Channel, which was right in the heart of the Wessex kingdom, although this Wessex-centric perspective was definitely assumed by the earliest author. He concluded his account, "Those were the first ships of the Danish men [i.e. Norse-speakers, probably actually from Hordaland in Norway] which sought out the land of the English race." This could very well be considered the beginning of the Viking Age, but 'viking' means raider, and this seems more like a simple murder by some merchants trying to avoid paying customs duties than any sort of organized raiding party.

So much for the historical record. We actually know there was a lot more traffic going on across the North Sea than these few lines state. The Viking Age definitely did not start with Scandinavians unexpectedly discovering Britain (despite The History Channel version of events). What instead seems to have happened was that this North Sea traffic grew increasingly violent, and although the kings of France and England certainly engaged in their fair share of violence, this particular violence slipped out of their control. The men of the church who had been supported by the royal violence of the preceding centuries and soon came to suffer as targets of this new violence outside of royal control left very vivid records of how bad things got for them. The 'Viking Age' is in some ways more about how and why monks started complaining about violence rather than about early medieval society suddenly gaining a new dimension of violence. (At the same time that three Norse ship crews reportedly killed an English reeve, Charlemagne was ravaging Saxony and hanging those he captured or deporting them into slavery—is this not also viking violence?)

Regarding the other side of Scandinavia, we're also starting to turn up evidence showing that Swedes were going abroad to raid perhaps as early as 650 or 700, that is, over a century before their Norwegian co-linguists set foot on Lindisfarne. But I don't think there was a straight line between these Swedish raids on 7th-century Estonia, Norwegian raids on the 8th-century English coast, and the 9th-century Danish armies of occupation. The Viking Age is really an umbrella term for a lot of different things, and it's probably impossible to identify a single event that triggered them all. So perhaps it's best in some ways to start the Viking Age with the texts about Lindisfarne that clearly show us the beginnings of a new perspective on early medieval violence.

TLDR There was a small raid of sorts on England in 787, a few years prior to the 793 Lindisfarne attack, and raiding had already been happening for generations in the Swedish Baltic. But whether either of these occurrences should be considered the start of the Viking Age really depends on what you define the Viking Age as. If you start digging into it, it's not at all as self-evident or as the one-sided as the authors of the Viking Age wanted it to appear.