Hello there!
As you said, the core rule book and the weapon section in d20pfsrd (http://www.d20pfsrd.com/equipment---final/weapons) list the sling as a proyectile weapon, here are my thoughts:
I would say that all kind of slings, the spear throwers and flask throwers are projectile weapons, but they are in the thrown section because they way you use them is very close to throwing something with your own hands(specially the sling). They are projectile weapons but they arent close enough to a bow or crossbow or firearm. "Assisted throwing" is a vague distinction since every proyectile weapon functions like a device that offers mechanical assistance and/or advantage at launching/throwing/firing objects, be it bolts, arrows, bullets, wathever. If you think enough, you will find that every projectile weapon is a device that will help you with something you can already do with your own hands.
- A spear thrower helps you throwing spears, harpoons and other spear like weapons or objects.
- Similarly, a flask thrower helps you at throwing splash weapons. There is a special type of crossbow too that lets you launch splash weapons.
- A bow or crossbow offers you a great way to launch specialized darts, adapted to the weapon itself.
- A sling and all of its variants offers you a better way to throw rock like amunition, you can even throw regular rocks and everything that is rock shaped and/or small enough to fit in the sling.
- Firearms are the epitome of doing harm "launching" something, there are plenty of specialized firearms that let you fire many types of stuff.
So, summing up, i think that slings and all of their relatives (except for some variants like the stitched sling which is used as a melee weapon until unstitched) and spear throwers should be treated as projectile weapons for feats and abilities unless some specific feat or ability tells you that you can aply it to projectile weapons in the thrown weapons fighter group.
An example of this is Startoss Style feat chain, from Pathfinder Player Companion: Weapon Master’s Handbook
Startoss Style (Combat, Style)
Your thrown weapons become more deadly.
Prerequisite(s): Dex 13, Point-Blank Shot, Weapon Focus with the chosen weapon.
Benefit(s): Choose one weapon from the thrown fighter weapon group. While using this style and the chosen weapon, you gain a bonus on damage rolls made with the weapon equal to 2 + 2 per style feat you possess that lists Startoss Style as a prerequisite (maximum +6 damage).
You cannot use this ability if you are carrying a weapon or a shield in your off hand (except for a buckler).
Special: In addition to the chosen weapon, a character with this feat and the weapon training (thrown) class feature can use Startoss Style with any thrown weapons that she wields in one hand.
It doesnt states specifically "you can use this feat with slings, all of its variants,spear throwers and flask throwers" but implies it because all of these are in the thrown weapon fighter group. I hope this helps you, sorry for the wall of text but it was needed to make a complete answer :).
EDIT: The rest of the feat chain would be a weird mechanic with the flask thrower, since lets you attack multiple targets by making the thing you throw bounce off of diferent targets. Unless you house rule it with something like "you decide in which creature make the splash weapon shatter and aply its effects, ending the bouncing there".