A lot of older vehicles allow fuel to run back. Just prime the key once before starting it if you think it's that. I had basically the exact same thing on a previous car but that was a 3.8L v6. Went 5 years with the car and 3 of which it did this. With that car it was the crank sensor, only found out because I knew the guy I sold it to and the sensor finally failed completely. Was never a code for it when I owned it but as soon as it was changed that car would fire up with half a crank.
2009 Ford Escape Limited - 3.0L, 6 speed auto, 4x4, 25/33 city/hwy imp mpg.
2017 F150 XLT-XTR - 5.0L, 4x4, 3.55's, E-locker, xtr chrome package, fx4 off-road package.
20/25 imp mpg