As a non-native speaker I felt my addition to what had already been said and written was needed. I also thought its premise was clear enough in saying that a word like invite depends on the context it's being used.
Fair enough invite is a verb, but learners shouldn't be surprised if they happen to hear or read "Thanks for your invite!".
As a teacher I think it's helpful to broaden learners views on different usages of the same word. That doesn't necessarily mean encouraging them to use ungrammatical expressions.
If anything I'd be delighted if a student of mine wrote on a quick note to a friend "Thanks for your invite!" and "Thank you for your invitation" to their friend's mum. That would mean not only were they smart enough to know the difference between a verb and a noun, but they brought it even to a higher level: they understood various pragmatic usages of the same word.
Yes, I don't want to argue with you over this. In Australia, our uptake of American (or British) novelties like this has a lag time of about 5 years. It's quite possible that there are enclaves throughout the world that use "invite" as a noun. But there remain many places in which it would be considered 'wrong'.
However, in no place at all in the English-speaking world is "an invitation" wrong. That's what (to me) makes it more correct. As another example, in AusE we don't generally use "gifting" as a verb. This is something that might or might not catch on.
There are some words which are in "the dictionary" simply because a minority of people use them that way, and the dictionaries are descriptive. If they're labelled
informal, that probably means that these usages won't be understood or used everywhere in the English-speaking world. That is, they are not 'informal' everywhere. In some places they might have a formal status, and in others, they might not even be recognised as being correct at all.
I would interpret
informal (in this case, and others I've notived) to mean "not used universally", rather than "used universally in an informal context".
I have noticed in learning Spanish that some words are used everywhere, and some are regional, or informal (but only in Mexico, for example). I'd like to think that the Spanish courses I'm doing are teaching me generally used words, unless thay are labelled as transient, trendy, regional, etc.
By the way, Australian novelty words take longer to reach the American dictionaries, but they occur too.