Is "When you play me" text considered an activated ability since you have to pay the cost of the card to play it?

No. "When you play me" is a Triggered Ability (specifically a "Play Effect"), not an Activated Ability.

The key distinction: paying the card's cost to play it is the cost of playing the card itself (following the normal six-step play process from Rule 349). An Activated Ability has its own cost followed by a colon ([Cost]: [Effect]), per Rule 377.1. Those are two separate things, even though both can involve spending resources.

When you play a unit with "When you play me…", the unit enters the battlefield immediately, and then that text goes onto the Chain as a triggered ability (Rule 383.4.a). Opponents can react to the trigger, but not to the unit entering play itself (FAQ #5735).