Starting trouble for me two weeks in a row

This is definitely a new guy problem, so I'll apologize for my ignorance in the first place.

I'm having trouble again getting my jeep started, but it seems strange to me. Here's the scenario:

-when the car has been off for 8 hours or more, it doesn't start on first try. Any other time no issues.
-battery has plenty of kick, it isn't a low battery I have had it tested and lights & radio no flickering or dimming.
-the engine will engage for about half a second, but doesn't actually turn over and stops
-if I wait for a minute or so it has been starting up after that just like normal
-starter was replaced last week, seems similar to the problem before the starter was replaced.
-I've double and triple checked the battery connections
-not sure if it's related, but when i start it, I push the brake and when it isn't starting, the brake pedal doesn't go down as far as

Thoughts?
Thanks for any ideas.

Side note: I have not left the lights on or doors open or anything that would drain the battery, plus if it was a dead battery I don't think it would start after a few minutes

When you say you checked the battery connections, did you also check ALL of the ground connections? A loose/dirty ground can cause weird stuff to happen.

I checked the connections on the battery, what other places should I check? I don't know where to even start.

There are a couple ground wires along the inside top of fenders each side....

I'll double check them. Thanks

I had troubles with my battery dying on me a while back. It would have enough power to turn on the lights and everything but not enough to start it. I got it narrowed down to an alternator that was running intermittently, which I replaced, but I also replaced the starter as an extra precaution. Haven't had a problem with it starting since.