I accepted the invitation and danced with Cassandra, but I didn't get the achievement.

Was this an Invitation or a Hastily Written Invitation?

It was an invitation.

The achievements only apply to the Hastily Written Invitations.

Oh, okay. I think it should say something about that on the achievement then. Because as it is now, it looks as if you need the normal invite.

how you get the normal invite

You can't any more. You can only get the Hastily Written Invitations now.

where do the invitations come from besides users shops?