What are the differences between the targeting perks?

Some items roll with various targeting and flinch perks on them. They seem to range from weapon specific to weapon types.

How do you know which is the better bonus between these perks?

For example,

Pulse Rifle Targeting
Improved target acquisition, accuracy, and aim-down-sights speed for Pulse Rifles.

Scatter Projectile Targeting
Improved target acquisition, accuracy, and aim-down-sights speed for Auto Rifles, SMGs, Pulse Rifles, Sidearms, and Fusion Rifles.

Kinetic Weapon Targeting
Slightly improved target acquisition, accuracy, and aim-down-sights speed for Kinetic weapons.

My guess is that the more specific the better, but I have no evidence as such.

Your assumption on specific being better than general is essentially correct. There is a reddit thread that contains a lot of information, calculations, and tests for perks and mods¹. As the reddit thread describes, there are three tiers related to the Targeting perk, as well as Reload Speeds and Unflinching.

• Tier 1: Slightly increases (or reduces) Y
• Tier 2: Increases (or reduces) Y
• Tier 3: Greatly increases (or reduces) Y

I scanned through the armor perk lists, and tier 1 is currently only applied to general weapon types (Kinetic, Energy, or Power) where as tiers 2 and 3 are applied to specific weapon types (Scout Rifle, Sniper Rifle, Rocket Launcher, etc...).

Below are the results of the testing regarding the tier levels for the respective perks. While some tests show some tiers having negligible differences, others show significant differences.

Targeting

All 3 tiers had extremely weak bonuses. The hitbox is slightly enlarged, but only a fractional rarely applicable amount, even with the tier 3 “enhanced” variation. The extra aim down sights speed was extremely hard to tell a difference in practicality (again, no exact/numbers testing was done).

Reload Speeds²

The difference is EXTREMELY noticeable. Tier 1 perks seem to be a 5-10% increase in reload speed, tier 2 seems around 15-20% increase, and tier 3 seems to be around a 30-35% increase in reload speed

Unflinching²

The tests were very interesting. Tier 1 had slight but hardly useful effects [...] Tier 2 prevents nearly all shake on the crosshair, but does not affect the crosshair displacement after taking damage or shield break. Tier 3 also prevents nearly all shake on the crosshair, and prevents most displacement on crosshair after taking damage, although displacement after shield break seems identical to tier 2.

¹ All tests were performed on PC
² Added for the sake of completion