I don't like the term connection, maybe indeed cause it doesn't seem to make a lot of sense to me. In therapy I had a aha-moment when learning that attachment styles refer to connection (or something). I really prefer the term attachment, I don't know why, but it's way more clear to me. Trust is also a good term.
I don't think connection means "common ground". You can connect with a baby or an animal and I wouldn't say you have common ground with them.