Since you asked about titles specifically, not parts, I’ll have to add the Wheelgun (made by Jakobs), which replaces the Revolver title in blue rarity (note: not blue-and-rarer, just blue). These titles are caused by no less than four different pistol barrels, though: Bandit, Dahl, Maliwan, and Tediore.
Other than that, it’s just the Torgue-barreled (a.k.a. quad-barrel) shotguns, i.e. oberkil! (Bandit), Development (Hyperion), Quad (Jakobs), Shotgun Supreme! (Tediore), and Ravager (Torgue). If we look at TPS as well, this list should be expanded to contain the Crowd-Pleaser (old Hyperion) and Bangstick (Scav). This is caused by a part restriction that prohibits the Torgue barrel from spawning on white and green shotguns.
There are similar part restrictions prohibiting a bunch of parts from appearing on white guns, making these only available on guns of at least green rarity. All accessories, non-default elements (except on Maliwan guns; default element is explosive for rocket launchers and Torgue guns, fire for TPS’s lasers and non-elemental for everything else) and scopes (except on sniper rifles and rocket launchers) are affected, but since titles are generally determined by the combination of gun manufacturer and barrel, all of these are pretty much irrelevant to the title. Okay, there are the Maliwan-barreled SMGs (and lasers, if we look at TPS) as well, where the element does play a role, but other than that… here are the barrels I know of that don’t come on white guns:
- the Bandit shotgun barrel (a.k.a. triple-barrel)
- the spinning Vladof assault rifle barrel (note that there is another Vladof barrel for assault rifles too)
- the Torgue assault rifle barrel (a.k.a. the grenadier / rocketeer barrel)
- the Jakobs assault rifle barrel (note that some of the titles it grants are not unique to it, most notably “Rifle” in the Jakobs case)
- the Jakobs pistol barrel
- the Vladof pistol barrel (note that this one cannot spawn on Jakobs pistols at all unless we’re talking about TPS’s glitch rarity)
With the Maliwan-barreled SMGs and lasers where the element changes the title it’s quite simple: the barrel itself can spawn in white rarity, but since non-default elements are green-and-rarer-only on non-Maliwan guns all of these special titles only appear starting at green rarity (except the Maliwan ones and fire beam lasers, of course).
Some more part restrictions of common guns: no Vladof or Tediore grips on Jakobs pistols, accessories are mandatory in purple and E-Tech rarities, no non-default elements on Jakobs and Torgue guns (you surely knew that already), elements are mandatory on Maliwan (duh) and E-Tech guns, E-Tech rarity only with E-Tech barrel (kind of obvious as well), and E-Tech barrels only in E-Tech rarity (except Hyperion pistols, which can technically come in purple rarity due to a bug: the E-Tech barrel overrides the rarity to E-Tech, but this bug makes Hyperion Darts and Spikers slightly more frequent and allows them to spawn without an element and/or with the gemstone material).
Edit: forgot one more obvious restriction: No E-Tech on Jakobs or Torgue.