Apparently, they are historically connected in the following way (or similar):
Vagy 'or' is a reduced form of the expression "az vagyon", 'the/that being/existence, that (thing which) is', with the implication of "the OTHER thing that is, ANOTHER thing that is", which was either implicitly understood or was also reduced/omitted in speech even earlier (cf. English or, which is a reduced cognate of "other", cff. German oder 'or' vs. ander '(an)other'). This would make it semantically similar to quite wordy English "other than that" (~ this thing other than that thing) expression.
Originally, and perhaps still archaically vagy-on is/was to be analyzed as the third person indicative singular of van, this being the reduced form of it. In other words, van 'is' is historically a reduced form of vagyon (cf. vagy-ok, vagy (itself presumably from vagy-sz), van (< vagy-on), vagy-unk, vagy-tok, vannak (< vagy-nak)).
Thus, vagy 'or' historically continues a phrasal reduction of the substantivized (verb used as a noun) form of the default form of 'BE' (stem vagy-), still extant in the form vagyon(-ok) 'things/goods, riches, wealth, substance, fortune'.
Hope you can sleep peacefully now, baratom :)
Do you really wanna know or are you just messing with me?!
The short story is they come from the same source form, which originally meant "form, shape, image, likeness, body" (cf. Dutch Lichaam 'body', German Leiche '(dead) body, corpse'), the semantic extension on the verb form being "of agreeable, conforming, pleasing form/likeness" (cf. Spanish hermosa 'beautiful' < Latin formosa (literally 'shapely', i.e. 'having [good/nice/pretty] shape') < forma 'shape' + -osa < IndoEuropean XY-wont-ya 'having XY'.
If something is LIKE something, it's like it's having the form of that thing (or 'of similar form/likeness'). If you LIKE something, it is of agreeable form/shape/likeness to you, you consider it to be of a fitting/good likeness (or some such).
They should be exactly the same. And they sound exactly the same to the Hungarian ear. That is, the difference you are hearing is not significant in the Hungarian language. Yes, it is very soft in this audio but it has no significance. It is just how it came out of the speaker's mouth in this case.
The short "a" in Hungarian is fairly far back in the mouth, so it can sound a bit like an "o" in other languages. Also, the "gy" is palatal, so if you're not fully hearing the stop and release of the consonant, you still might be hearing the movement of the tongue to a palatal position, which is where /j/ is pronounced ... so hearing it as "voi" is actually pretty understandable.
The thing is this is an actual voice recording, not standalone words text-to-speeched. I can only speak in the name of Polish and Romanian courses... I'm not really happy with the Hungarian course but this choice with audio feels right actually. You can slow text-to-speech down the way you feel like - but it's just not legit. At the end of the day, pronunciation you hear here is what you'll have to understand and try to mimic.
The word "not" is a negative adverb. "Yes" is an interjection, as are "Wow!" or "Ouch! " So the word "no" in the question "No, or yes?" would be an interjection of negation, like "Ewwwww!" (in "my" English). Consider: "Hooray, or boooooo?" - "Igen, vagy nem? [And no! "nem" is not a negative quantifier in this context. (Or could it possibly be so by implication - "Magyarul"?]
I have a "grammar-contrary", but creative, friend who, regularly, does (or, at least, he tries to) perempt and pre-empt my proposed or continuing course of action with that obnoxious expression: "Not!" But no! I am not the oppositional one. Someone please correct me if I am wrong!