QUICK NOTE BEFORE THE LIST: The weapons on the DLC outfits included in the game DO count for the achievement. I was using the Agent 47 outfit (which comes pre-equipped with the Silver Baller Pistol) when my achievement popped. These outfits can be bought in the Night Market at the vendor marked in brown/copper. They're expensive, but you earn money quickly in this game so obtaining these outfits should be no problem.
QUICK NOTE 2: Killing civilians DOES count towards this achievement. Again, using the Agent 47 outfit, I shot and killed a civilian, and achievement popped.
FINAL NOTE: There are 16 guns you can obtain before the end of the main story. Keeping an eye out for guns to use, and using the 3 DLC guns, you should easily get this achievement by mid-game.
Now, on to the list of where to find the guns. Credit to the Sleeping Dogs Wiki for most of this information, I've annotated the information to make it relevant to the DE, and to add tips as needed. Minor spoilers included below.
'DLC' Weapons (You can get these thanks to this being the Definitive Edition)
Rico Rodriguez's Signature Revolver
Comes with Rico Rodriguez's Outfit.
Silver Baller Pistol
Comes with Agent 47's Outfit, a silenced firearm.
FR-27 Rifle
Comes with Sarif Industries Armor
Handguns
9mm Pistol (Heckler & Koch USP)
Found on HKPD police officers. Police will begin to use them at a Heat level of 2 if the player draws out a gun or gets into a vehicle. Also found fairly commonly on gun weilding thugs, especially towards the beginning of the game.
.45cal Pistol (Heckler & Koch USP )
Only used in the mission 'Meet the New Boss' by Lee's men. The fight continues from the club out to the street, so you have time to clear the club before you pick one up.
.45cal Pistol w/ Taclight (Heckler & Koch USP)
Again, common among armed Triads, though these tend to show up a bit later in the game.
Silver .50cal Pistol (Desert Eagle)
First being used in the mission 'Meet the New Boss'. It does not appear in any future mission. Again, the fight continues from the club out to the street, so you have time to clear the club before you pick one up.
Golden .50cal Handgun (Desert Eagle)
Available partially during all mission playthroughs of "Meet the New Boss " and 'Big Smile Lee', and is available in Safehouses after the main story. You'll start the shootout in "Meet the New Boss" with this equipped, so this is the easiest place to get this one.
Submachine Guns
.45cal SMG w/ Taclight (Heckler & Koch UMP9)
Common from armed Triads and HKPD SWAT. You'll probably pick this one up without even knowing it (I found a bunch in one of the camera hack areas accessible by boat).
Golden .45cal SMG (Heckler & Koch UMP9)
Only available during the 'Triad Highway' mission (part of the 'Triad Enforcer Pack' DLC, included free in the Definitive Edition, so you have it).
Machine Pistol (Heckler & Koch MP7)
Introduced to by Roland Ho during the mission 'Uncle Po ' early in the game. Your mainstay weapon for taking down automobiles, so you'll get this one no problem.
Shotguns
Anti-Riot Shotgun w/ Taclight (Hawk Semi-Auto)
First used in the mission 'Civil Discord'. Looks like your normal shotgun but has a light on the end.
Pump Shotgun (Hawk Type 97-1)
Common from armed Triads, HKPD PTU, or police car trunks with Trunk Key Cop Upgrade (unlock this at the far end of the lower Police XP track. Police XP comes quick, so this shouldn't take long).
Assault Rifles
Assault Rifle (QBZ-95)
Available in the mission 'Civil Discord' in the Golden Koi restaurant lobby, about 2/3 of the way through the main story. You'll spot it easily..
Assault Rifle w/ Taclight (QBZ-95)
First seen in 'The Wedding', and agin in 'Mrs. Chu's Revenge' although players may not have a full chance to use it in 'The Wedding' and 'Mrs. Chu's Revenge' since they only appear near the end of the missions. Your best bet to get it early (in 'The Wedding') is to take out all the thugs but one, then search the corpses for this weapon (it will most likely be on a thug right by the exit door). If you can't find it, Restart Checkpoint, as the last thug is holding it. When you go through the exit to find Uncle Po, this gun will disappear so make sure not to leave the wedding hall until you've used it.
Assault Rifle w/ Grenade Launcher (fictional L85A2 & TAR-21 w/M320)
Available during the missions 'The Funeral' and 'Civil Discord'. Their almost impossible to miss as you'll need the grenade launcher in 'The Funeral'.
Addendum on the Assault Rifle w/ Grenade Launcher (Thanks Allgorhythm): Note that the Tactical Soldier Pack DLC outfit comes with the same Assault Rifle w/ Grenade Launcher that you get much later in The Funeral. So, not only can you get the weapon to count very early as you work to this achievement, but you can also walk around town blowing up cars to work on your gold stat for destroying vehicles. The Tactical Soldier Pack also reduces gunfire damage so you can combine your activities and work on your gold stat for rogue cop as well.