Opening this up again because I'm having similar issues. So far i've replaced my battery (9/3/19) and my starter with a used but healthy starter (9/22/19).
Here's the deal - if I put my key in the ignition, wait a second or two to let the ceremonies finish, then try starting it I will either A. hear a "click" or B. my engine cranks once or twice then cuts off.
Now, if I put my key in the ignition quickly and turn the key quickly without letting any ceremonies finish, I get about 10 engine cranks and then cuts off. This is a repeatable phenomenon.
Brand new battery, healthy starter, ~12.5 volts off, 14+ volts running, and I cleaned/tightened all connections at battery, starter solenoid, and battery ground. No one I know has any ideas.. can anybody take any guesses? EDIT: Forgot to add, push starting did not work after 4 tries.