Neither of those are the reason, speaking from experience both as a family tech support and as a web dev. It’s because everyone recognized the big “e” icon. My brother got my dad a Mac and he already knew the main browser was Safari, but he still called me (knowing I had used Macs for years longer) and just asked:

1. What the hell is Safari (no kidding he didn’t even say hello).

2. Can I use the Internet Explorer?

Edge now (correctly) accounts for this by... having a big “e”-looking icon and basically otherwise being a skin over Chromium.

I think you're confusing cause and effect here.

Yes, the `e` became synonymous with the Internet for a lot of users, but that's only because of the two reasons listed. Not the other way around.