What qualifies as a "precision loadout weapon"?
A "Precision Loadout Weapon" is a precision rifle-type weapon that you can spawn with.
There are 4 weapons that meet this criteria: DMR, Battle Rifle, Covenant Carbine and Light Rifle.
If you want to quickly complete this challenge, I would recommend playing SWAT, since you have to start with a precision weapon and enemies die with a single headshot.
Also, keep in mind if the playlist has you spawn with a precision weapon that is normally a pick-up weapon, this still counts. For example, in Team Snipers you spawn with a Sniper Rifle; this does count towards these type of challenges.
Here is a forum thread about challenges that touches on this topic: Xbox360 Achievements - Halo 4 Challenges Thread