A creature that is equipped is considered modified no matter who controls the Equipment that's attached to it.

The value of X in Golden-Tail Trainer's last ability is determined only as the ability resolves.

An Aura controlled by another player does not cause a creature you control to be modified.

All costs to cast a spell are locked in before mana abilities are activated and those costs are paid. If Golden-Tail Trainer's power changes after costs are locked in, or if Golden-Tail Trainer leaves the battlefield while you are activating mana abilities or paying those costs, the cost reduction effect won't be affected and the total cost to cast the spell won't change.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying, add any cost increases, then apply any cost reductions. The mana value of the spell remains unchanged, no matter what the total cost to cast it was.

A creature with a counter on it is considered modified no matter what kind of counter it is or which player put it on that creature.