The text in the card saying “when you play me” when can you consider a card being played?

A card is considered "played" once the act of playing it has been completed by the card resolving — not when you announce it or put it on the Chain (Rule 419.4.a).

So for a unit: you put it on the Chain, it resolves and enters the battlefield, and only then does "When you play me" go on the Chain as a triggered ability. If the card is countered before it resolves, the trigger never happens (419.4.a.1). Triggered abilities that check "play" mean "resolve"; non-triggered checks (like Battering Ram's cost reduction) mean "finalize."