"Register": I immediately know what it means and it takes tenths of a second for me to make the decision to click (or not to).
"Sign up": despite having a CEFR C2 level of English and using it at work every day, I still need to think for a few seconds if this is a registration or a login (cf. "sign in").
For a native speaker I suppose the latter is (marginally?) easier, but for a non-native, it's much harder, not even close.
Your examples are very illustrative, but I would argue that both phrases suffer from being too generic. I wish that there wasn’t such a push for “one word” CTAs. Examples of alternatives I’d advocate for depending on context:
Obviate has several cognates in romance languages, so it is straightforward.
Now, the phrasal verbs that are so obvious to a native speaker, these are the real head-scratchers.
Or street slang. It may sound like dumbing it down to you, but it actually adds complexity for a non-native.