The second ability represented by the "partner with [name]" keyword modifies the rules for deck construction in the Commander variant and has no function outside of that variant. If a legendary creature card with "partner with [name]" is designated as your commander, the named legendary creature card can also be designated as your commander.

Once a creature's training ability has triggered, destroying the other attacking creature or reducing its power won't stop the creature with training from getting a +1/+1 counter.

"Partner with [name]" represents two abilities. The first is a triggered ability: "When this permanent enters the battlefield, target player may search their library for a card named [name], reveal it, put it into their hand, then shuffle."

A creature's training ability triggers only when both that creature and a creature with greater power are declared as attackers. Increasing a creature's power after attackers are declared won't cause a training ability to trigger.

Note that the target player searches their library (which may be affected by effects such as that of Stranglehold) and that the card they find is revealed, even though these words aren't included in the ability's reminder text.