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.
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).
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
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