What actually defines Fallout canon, and why should it be Bethesda?

Solution 1:

In fiction, canon is the material accepted as officially part of the story in an individual universe of that story.

Source: Wikipedia

In other words, canon is every event that "really" happened according to a given work's story. In this case, the work is a video game of the Fallout series. What is or isn't canon can differ from game to game.

For Fallout 1, everything that is told within the game is canon. After all, Fallout 1 represents the starting point of the series, and there is no other material that can influence the story^*.

But since the Fallout series has branching storylines, some events may be mutually exclusive. For example, the Dweller can't have saved the Vault if they failed to save the Vault (obviously) and vice versa. And once an NPC has been killed, any event that can only happen if they're still alive automatically becomes non-canon.

However, canonicity usually isn't relevant within the same piece of work, because it is too obvious. If the hero set out to fight the dragon, then they set out to fight the dragon. This is why canonicity is usually only discussed when additional works exist, such as spinoffs, prequels, sequels, interquels, etc.

So when we look at Fallout 2, the sequel to Fallout 1, obviously something must have happened during Fallout 1 that led to the story of Fallout 2. That "something" is canon. Did the Dweller save the Vault, or did they not? Did they stop the Supermutant invasion, or did they not? Not every event that can happen in Fallout 1 necessarily happened in Fallout 2. That is why some events of Fallout 1 are canon in Fallout 2, and others are not.

So to answer your question: "Canon" is officially determined by whichever game you're playing.

And yes, it is quite possible for different games of the same series to have different canons. One of the best examples being the The Legend of Zelda series, where 3 different (mutually exclusive) events in Ocarina of Time can be either canon or non-canon, depending on whether you're playing A Link to the Past (the hero lost), Majora's Mask (the hero won as a child), or The Wind Waker (the hero won as an adult).

As for your other question: "Why should it be Bethesda?"

Because Bethesda owns the rights to the Fallout series, and as such are the only ones to be (legally, and therefore also officially) allowed to create any more pieces of work set in the Fallout universe.

^*: That is until Fallout 76 was released, which is set before the events of Fallout 1.