Need to get yourself a test light. You can find where the power is failing to reach. Start signal is reaching the solenoid, the solenoid is kicking, I don't think 12 volts is reaching the starter motor, but it could be reaching the starter and it's just not turning. You checked the battery connections for being tight, you need to pull each connection from the battery to the starter to check for corrosion. The negative connection needs to be checked at both ends as well, but everything is staying lit up, don't think that's it. Get that test light, a simple diagnostic tool that needs to be in every tool kit..