Tip from a non-native English speaker: if you want your website to be more friendly to an international audience, don't use the terms "sign in" and "sign up", use more distinct terms (like "login" and "register") instead. Phrasal verbs, in general, are difficult to speakers of languages that don't have them, especially when the same verb has different meanings depending on the added preposition. Someone with a basic/…
Afaik, you "sign up" for something, like a newsletter, or health insurance. You sign into a hotel, ie leave your signature at the check-in (or sign-in) desk. Btw, also not a native speaker.
By the way, this is why an English speaker would say "sign in to a website" but not "check in to a website". Check in is generally a non reoccurring action.