I wanted to see if dealing damage or taking damage (for small hits) affected the limit break bar, and after running a number of trials I believe the answer is no.
The primary contributors to the Limit Break bar seem to be time and encounter mechanics.
Note that you still gain a small charge when healing someone at or below 10% health or when a defensive cooldown mitigates otherwise fatal damage, but both of these will comprise a fairly small amount of your Limit Break charge.
For four-man content, I'm relying on the data posted by reddit user freakngmayhem:
I began with a party of four level 60 characters, SCH, SMN, WHM, BLM, a level 60 striking dummy, and a stopwatch.
I used a countdown timer to tell the party when to begin. We all casted Virus on the striking dummy to enter combat, and then sat there doing nothing until the limit break bar filled. We stood at range, and did not auto-attack the dummy or perform any actions. It took 6 minutes and 40 seconds.
We disengaged again, pulled again, and this time did as much damage as we could to the dummy. After a while, I started just spamming AOE heals. It took 6 minutes and 40 seconds.
I disbanded the party, and went into Expert Roulette by myself. (BLM, PLD, SCH, SMN.) I kept my stopwatch at hand. Every time we entered combat, I pressed start, and then ended it when the fight was over. I performed all of my normal actions, including the occasional sprint. Nobody was spiked to low health, was healed up from low health, or interrupted any attacks -- no "fwip" sounds. Midway through the first boss, we filled our limit break bar. It took 6 minutes and 40 seconds.
This test was mainly done to try and disprove the myth that sprint builds limit break, but it's also noteworthy that the amount of time to charge the bar was identical between the attempt where everyone was doing full-on dps to a combat dummy and the attempt where everyone sat around afk.
While I think this answers my question, there was still some uncertainty about whether or not this applied to 8-man groups.
I unfortunately didn't have eight people to sit around a combat dummy for half an hour. So the next best thing was to queue for a bunch of Duty Finder runs.
I picked Thordan (Normal) for my test fight. Thordan is a level 60 fight, so no level penalty. Queuing via Duty Finder meant I would not be penalized for duplicate jobs in the party. And Thordan has virtually no adds to kill (comet circles made no noticeable impact) nor any skills to interrupt, which allowed me to focus on time vs damage dealt/taken.
Across 20 runs, the time it took to charge a segment of the limit break bar was consistently 2 minutes and 15 seconds. (Or 6:45 to get a full limit break 3.)
Now, I couldn't control for the gear of my party, but I was able to note down how geared everybody was. On both the runs where I was with fully-melded ilvl 220+ dps and four new-to-60 sub-170 dps, the time it took to charge the bar was exactly the same. Given the power curve on gear upgrades, if damage dealt or number of crits mattered, you would assume that the fully-decked out groups could charge the bar faster.
Likewise, the bar charging rate was constant between heavily geared tanks and fresh-to-60 tanks, so I do not believe damage taken is a factor. Furthermore, there is a full minute of special effects and extremely light raid damage (<1000 per hit) where Thordan and his knights charge up his ultimate attack, and the limit break bar charged at the same rate as it did the rest of the fight.
Still, there is not a lot of damage flying around on the Thordan fight, so I wanted to see if the rate would be the same during something more punishing. So I queued up for a number of Alexander Turn 8 runs, which is chock-full of both raid-wide damage and tank busters. Even with the increased raid damage, I found that the time to charge a limit break segment was still 2 minutes and 15 seconds.
However, interestingly enough the rate of charge during phase two (where you fight the pairs of robots from Alex 6) is significantly slower. The bar charged at the regular rate during both Phase 1 (measured by the time it took to go from 0 to 1 segment) and Phase 3 (measured by the time it took to go from 2 to 3 segments). I don't have the exact rate of the slower charge yet (I suspect it charges at or a little more than half-speed), but I believe this falls under the category of "encounter mechanics".
Cursory experimentation with other level 60 boss battles also suggests that some fights speed up or slow down the limit break charge during certain phases of the fight. E.g. Ravana Ex will speed it up so you have limit break 2 ready before Chandrahas. But I think these variations are similar to Ifrit's Nails, where the encounter is tuned so an average group at-content will have limit break 3 available near the end of the fight.