Will either crank really slowly, crank once/twice then stop cranking?

Starting about three weeks ago, my vehicle is having issues starting in the cold or restarting after it's been running for a short amount of time (such as driving for 5 minutes for a trip to the grocery store, then starting again to leave).

The motor will either crank really slowly, crank once/twice then stop cranking, or not crank at all. The issue is more prevalent when it's cold outside (below 30 degrees). Usually, I'm able to wait a minute or two then try starting the motor again. I've also found that if I turn the key to the 'accessory' position then try to start, this issue doesn't occur as often.

Some notes on the vehicle:
-Has only been to Honda dealers since new for service
-Recently had a 100k mile service performed which included new spark plugs, serpentine belt, transmission fluid change, coolant change, brake fluid change, and oil change

It was at a local Honda dealer last weekend where they ran a full charging system check. The battery, alternator, and rest of the electrical is great. They fault the starter. Not wanting to spend $700 on a new starter installed at a Honda dealer (it is almost a 10 yr old vehicle), I went to a local Firestone where they fault either the starter or the master ignition switch. Firestone felt leery about replacing the starter when they couldn't replicate the issue I'm having.

The question is, could it possibly be anything else other than the starter, if it is the starter, would a re-manufactured starter be as good as a new one from Honda. It's a big difference in price, and for a vehicle this old I don't think spending $700 on a starter is worth it.