Yes, the reinforcement logic is largely identical to any other planet. The few key differences:
1. If the homeworld is NOT on alert and all the other planets on alert are full or nearly so, then the homeworld gets a slightly higher priority of reinforcements compared to just other random reinforcements.
Result: If there were 10 planets on alert for a long, long time and they all filled up, the rest of the galaxy would be getting the overflow reinforcement points at random; but the AI homeworlds would get slightly more of them. That could easily lead to the larger numbers you saw in your other game.
2. The homeworld generally, but not always, draws down 2x the reinforcement points and draws down them first when it's on alert and isn't full.
Result: If a reinforcement cycle happens soon after you get to a homeworld, generally the number of ships can skyrocket abnormally quickly. If in a game you accidentally telegraph your position before you mean to (and alert the AI homeworld), and don't notice, then by the time you actually look at the homeworld it could be 2-4 times larger than it otherwise would have been, even if it's only been 5-10 minutes of such telegraphing.
In general the AI Progress doesn't have anything to do with how many reinforcements you'll find at the homeworld when you first get there, if it hasn't been on alert. I mean, there is some trickle-down effect from that in that each reinforcement event is larger at higher AIP, and thus you're more likely to get to overflowing on other planets that would then trickle over to the homeworlds, but there are a lot of factors that are more predictive than the AIP, as I noted in #1 and #2 above.
The AIP
does affect how quickly the AI homeworlds can ramp up once they do go on alert, though. So it still makes it a tough fight, especially if the homeworlds take a while to fall. But that's why so many people advocate lightning rushes on the AI homeworlds so strongly -- this is a time when being able to direct strong, ongoing force really can save you a lot of time and ships!