Sending my whiteflame protector unit from a battlefield that has a enemy unit, showdown trigger, does that consider my whiteflame protector being played? Do the ability “when you play me, give a unit +8 might” takes effect?

No. Moving/sending Whiteflame Protector to an occupied battlefield starts a Showdown, but it is not "playing" the unit, so its "when you play me" ability does not trigger (Rule 383.4.a).

"Play effects" only trigger when the permanent is actually played to the board from a non-board zone (hand, champion zone, etc.). A unit already on the board being moved by an effect is a Move, not a play — Play Effects trigger on the play action itself, not on entering a battlefield. So Whiteflame Protector would enter the Showdown with its printed 8 Might, no +8 buff from its own play trigger.