Guild Wars 2

No, it works on anyone (randomly) in a radius of 600. Up to five allies regardless of being in a party or not.

I think it triggers on the effect of Virtue of Resolve on yourself.

Dunno about the short cooldown anymore, but you can easily test this by yourself by getting the trait and standing right next to a couple people.

See the wiki page for Battle Presence:

• Has a radius of 600.
• Allies’ healing from this trait is not affected by Absolute Resolution.
• Works even when the user’s Virtue of Resolve is on cooldown.
• When triggered, it will tick three times on up to five allies while going into a short cooldown afterwards before it is reapplied.
• Allies effected by Battle Presence have Virtue of Resolve’s passive effect shown on their effects bar.

You will always see which characters are affected by each tick of Battle Presence even when everyone around you is at full health as it will show an annoying “Immune” above them each time…