Does the point of sale system ask for a tip? Is there a check with a tip line on it that you sign? Is there a prominent tip jar? Then: tip (or do business elsewhere). Else: don't.
You're missing the point. The existence of that line or that jar will change whether tipping exists, even though your relationship with that person hasn't changed at all.
So maybe it's not really a three way relationship in any of these situations.
So maybe it's not really a three way relationship in any of these situations.