2002 Toyota Corolla Starting Problem

Had new battery 750CCAinstalled $100.00, all cables and belts checked -14.4 volts at battery when running! Had starter/selinoid rebuilt -$200.00
No check engine lights, plug pulled looked fine!
Taking two turns of key to start car that rolls over for a few turns then DEAD silence best way to describe it. Turn key back to off, wait a few seconds turn key back to start to another few roll overs then usually barley starts with what I would call a empty vacumm sound followed by barley hearing the car is running and no sputtering or missing once started. Seems like first moring start is when it does this more than mid dayStarted back mid Septand thought it was battery. Heard there is no ignition relay for this car but a modual behind key. What is your best guess with HUGE THANKS in advance and show a scematic of starter system for me to see again with thanks-nanccinut