A solenoid is a device that converts electrical energy into linear motion. Some starters have a built-in solenoid that causes them to engage, but typically the term "starter solenoid" refers to a solenoid that closes a high current switch to supply power to the starter. Some vehicles also have a starter relay (a relay itself being a specific sort of small solenoid switch), which provides power to run the starter solenoid, as some require north of 10A, and that alone requires a fairly heavy switch.
Back to the point... If the bikes crank over fine on a charger, that means you're just shy on power. Sounds like a couple of amps on the bike with the big battery, and a good few on the one with the small battery. You can either replace the solenoids with some that require less current or replace the batteries with some that provide more current. Seeing as solenoids are cheaper than batteries, it's no fun trying to fit a bigger battery in a bike, and adding amps means increasing the electrical load overall with no real benefit, I'd recommend replacing the solenoids.
Again, Rick's makes them. Thirty to forty bucks. Get them (probably a special order) at pretty much any bike shop. Everybody gets stuff through Parts Unlimited and/or WPS, and both have them.
You might look into wiring in starter relays as well, especially on the bike with the small battery. Use an SPDT relay to switch the headlight off when the starter is cranking. If you're running the headlight, that's anywhere from 3 to 5 A you're giving up that could be powering the starter.