It is inaccurate to say that Gilbert Baker created the first rainbow flag. I realize that many sources do say this, but as you note, it is imprecise. His own website says he created the "LGBTQ rainbow flag," which is accurate.

Beyond this point is really a sociological and/or linguistic question rather than historic, but I'll pontificate a little, if the mods don't mind.

So, how did people come to just call it the "rainbow flag" then? It's just popularity, really. While Wikipedia has collected several other rainbow flags besides Baker's - dating to the 16th century! - none of them are as globally relevant to such a large segment of the population. While the peace flag is arguably universally relevant, it is simply not as well-known. While there was a campaign for peace in Italy that utilized the flag, it just doesn't rise to the same level.