But “invite” (accent on the first syllable) as a noun meaning “invitation” is less acceptable: “I got an invite to my ex-wife”s wedding.” Though this form has become extremely popular, even in fairly formal contexts, it is safer to use the traditional “invitation.”

Which is correct invite or invitation?

Invitation is the more accepted noun to use. Using invite as a noun is informal.

Is it Thank you for the invite or invitation?

I am sure, though, that in more formal situations, most Americans prefer to thank their hosts for the 'invitation'.