Exchanging harm is a move like any other. It happens when the fiction triggers it: “When you exchange harm, …”. This can trigger because another move inherently causes such a fictional situation and calls attention to it, or during the Conversation with no other move involved.
Keep in mind that moves aren’t push-buttons: that can’t be chosen or declared by players. Players who want to trigger a move have to take fictional actions that trip its trigger naturally. As AW says, “to do it, do it”. Exchanging harm is triggered when the situation is that harm is being given and taken due to the situation and the PC’s actions making that true. A player can’t “use” it to just do damage.
How to just do damage, as a player
Shoot someone. Hit someone. If not that easy because they’re too far for a sure shot, shoot at someone — the MC will tell you what happens. (Which is to say, you-the-MC will either see a move has been triggered, or if not this is the MC move trigger “look to you” and so you’ll pick an MC move that follows, honestly, from the fiction and your Agenda and Principles.)
If they’re also hitting or shooting you back and y’all have sure hits, you’ll exchange harm (and exchanging harm will trigger). If they’re trying to hurt you back but it’s not a sure thing maybe you’re doing something under fire and there’s more to see before you really take your shot. Maybe something else, something good or bad. Apocalypse World is a place full of things we didn’t plan for.
If you shoot, then how it all goes down depends. You might just hit ’em and they’re gone, because you suckered someone. You might miss and draw fire that you’ll be asked what you do about it. You might wound them or might scare them, might pin them down or run them off. You might hit a friendly or damage something someone or you care about, your gun might jam, you might have someone pop up beside you with a gun to your skull just as you lay finger to trigger. The Maelstrom might wisk you and the flying bullet into a psychic dream and make you discuss your relationship with your awoken ammunition. Depends on the situation. Set up your situation with care, if you can, and if you can’t, be situationally aware because who knows what comes next.
Any way it goes, it always goes like this: say what you do, talk through the fiction as it unfolds, and touch the mechanics only when and how the game says to. Usually this means leave ’em alone and just do stuff until a move’s trigger matches what’s already happening, then do what it says.