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The power-up effect is very specific. Getting a +1/+1 counter, even getting a +1/+1 bonus that isn't from a specifically worder "power-up" effect isn't technically a "power-up", and therefore, Lost City does not apply.
If you're not sure what I mean by that, look at the card "Tamaran". Compare that with Cassandra Cain <> Batgirl and Bastion, whose effects grant the same amount of a boost as a bonus, but their effects are not counted as power-up effects.
To make it easier for you, since you're not doing Juvenile Delinquents (Brotherhood / Titans), think of it like this. Lost City only works when you discard a character card of the same name as one you control. The bonus only lasts for one attack.
Any other time something such as equipment or Savage Land gives you a bonus, though it's easy to describe it as "powering up the character", that's technically not correct. "Giving a bonus" or "pwning your face" are acceptable substitutes. ^_^