Short story
Pocketed is better for carrying capacity. Lighter Build helps with stealth and AP, but not nearly as much with carrying capacity.
Long Story
The Nukapedia page Fallout 4 Armor Mods has weights and effects listed for various pieces of armor. As of writing this, it was not completely filled out for all items. It does have all the
information for metal chest armor, so let's look at that as an example.
Polished Metal Chest Armor weighs 13 lbs.
Lighter Build reduces the weight by 3.2 pounds.
Pocketed increases the weight by 0.8 pounds, but adds 10 carry capacity, for a net gain of 9.2 pounds.
Ultra-Light Build reduces the weight by 6.4 pounds.
Deep Pocketed increases the weight by 0.8 pounds, but adds 20 carry capacity, for a net gain of 19.2 pounds.
So, clearly, pocketed is much better for carry capacity in this case.
Scanning through the other armor, the exact difference between the improvement provided by the pocketed mods and the improvement provided by the lighter build mods varies, but in all cases it appears that the pocketed mods are significantly better for carry weight. This applies to both chest pieces and limb pieces.
Perk Efficiency
You asked whether it would be better to get a higher rank in the Armorer perk or just put more points in strength. That's a pretty complex question, that will depend a lot on subjective preferences. This is because Armorer unlocks a lot more than just the Deep-Pocketed mods, and Strength does more than just increase your carrying capacity.
That being said, if you just want to maximize carrying capacity, you should invest in Armorer.
It looks like Deep-Pocketed only requires Armorer 2 on most, if not all, types of armor. Two points of strength will only gain you 20 lbs of carrying capacity. Putting those two points into Armorer instead allows you to put Deep-Pocketed on all your equipment, gaining you more than 20 lbs on any set of armor.
The closest I found to a workaround for this issue was to reload an old save, from before I had manually attacked the Prydwen. After doing that, I found the Underground Undercover quest progressed correctly with the following in-game state:
- Glory is alive.
- BoS members on the airship are alive.
- BoS members on the ground at the airport are alive.
- Paladin Danse is dead.
- The 'Underground Undercover' quest wants me to warn Desdemona.
- I haven't progressed the 'End of the Line' quest past the point where it wants me to talk to Father.
After that my game worked as it should (got to fight off the BoS attack, got the mission to attack the Prydwen, looted Final Judgement off of Maxson's corpse, was able to warn Desdemona again after talking to Father, etc.).
Another interesting thing I found was that if you progress Underground Undercover to the point where it wants you to clear the control room, you can instead turn around and take the elevator back into the Institute proper and pick a fight with everyone without getting banished. Among other things, that lets you take out members of the directorate and other named Institute NPC's who disappear if you just keep progressing Underground Undercover.
Best Answer
Well, as it turns out, when entering Power Armour, your strength is SET to 10. I.e. Power Armour gives you maximum strength, affecting carrying capacity as normal. Therefore by having 10 strength you do not benefit from that particular Power Armour bonus.
As a side note - it should be noted that if you have pocketed armour, which increases your carry weight, you do not benefit from that when IN power armour. In other words, if you rely on greater carrying capacity from Power Armour you should consider these points.