Weird starting issue

[shepd]

So I'm having an odd issue starting my car. I replaced the battery recently with a brand new one, as the old one was definitely toast. However, the issue I'm describing was happening before the replacement and still after it.

Basically, I turn the key to start the car, and the starter engages and turns over the engine for a moment or two. Then it stops for a moment. And then it tries again. This does not occur at all times, but perhaps 1 in 3 tries. Sometimes when it's like this it cannot start the car at all, and I must repeat putting the key in start, at which point it usually does start without trouble.

I've had this this issue mostly after I've been driving, even if it is just for a short period. I know sometimes something like this can be due to a bad fuel pump and lots of heat, but then, wouldn't the starter keep trying instead of just stopping? It's also cold as heck here and it still happens even on a short drive from being left to freeze overnight.

I've had my door open while this issue occurs and I can hear something straining, I'm guessing the starter?

When starters go bad, is this a typical way they die? If it's toast, any remove/replace guide here? I do have the factory service manual, so that's at least something. :)