BEFORE YOU DIDDLE THE BATTERY CONNECTIONS READ THIS.........
The battery is to be connected to the engine with the - terminal connected to the engine block via the black lead. This may be through the starter housing and connection to the block is via that or a jumper wire but - is negative, ground, return, zero volts and anything else you want to add.
+ is the Red lead and it goes from the battery + lead (12v source voltage) usually to the input terminal of the starting solenoid which supplies voltage through the solenoid (when energized) to the starter, but is used as a junction point for 12v distribution throughout the engine and remotes.
If you reverse your battery terminals from what I posted, even for ? of a split second you will burn up the negative diodes in your battery charging circuit in the engine. So look before you leap. If your problem IS that the lines are reversed, your rectifier/voltage regulator doesn't work any longer!
Follow David's mention of the 2 finger test on the bendix with the engine off and making no attempt to start it. Spinning fast may seem like it is working ok, but for the Bendix to work when the solenoid closes, you need about 150 amperes to SLAM the Bendix into the flywheel.
Check your voltage across the starter when you attempt to start it; one lead on the ⅜ stud on the side of the starter and the other on the starter case....metallic band if composite housing. When you hit the starter you need to have at least 10v present while delivering the necessary current. Course when the Bendix misses connecting with the flywheel, you aren't using anywhere near 150 amps so you ought to read close to the full battery voltage.