Here’s what the 6th edition rulebook says about matches (chapter 8, section 4):
Under this definition, Alex was correct to code the NH-ME team match as a match. I don’t think the OP’s match-play club format would be considered a match any more than a double-round Swiss would be considered a match, because each player plays more than one opponent and isn’t playing “all or substantially all” of his/her games against the same opponent.
And thanks to the moderators for splitting this thread into two threads, each of them interesting on its own. I was about to suggest such a split, but when I turned on my computer I saw that it had already been done. Good work!