Hello Harald, welcome to this forum. :)
invitation is a noun; invite is a verb
In the last few years, many have been using "invite" as a noun, at least informally.
I would suggest that if this is a work environment, even casual, you'd be better to stick to invitation. It's just the safer route IMHO, especially if you are not a native speaker of English (not that there's anything wrong with being a non-NES of course, but sometimes it can come across as odd - like trying too hard). Maybe I am just old-fashioned, or just old :)
On the other hand, if you were responding to a friend asking you to join him for a beer, sure: thanks for the invite!
Thank you for the invitation, XXX. I'd be (more than) happy to participate.