First off, a 2003 is a DI, not an IDI. IDI is the motor before the powerstroke. What you describe is the classic bad starter. Yes it appears to spin fast, faster than a gasser. BUt as most find, if its not 150rpm or more, it will not start. In your case, of course the glow plugs help , as a warm engine starts easier. The fact you sense a cylinder or two trying to fire, shows its not spinning fast enough. Go to the parts store and buck up for a denso starter. Its huge, barely fits, some like me even had to take a quarter inch off the cross member to fit it in there. Well worth the money. You will immediately see the spin speed difference. OR just get a stock one. Try it before you return your core. Bet it fixes the issue. If not return it , no harm done. But your description totally points to a slow starter. Also, DO NOT FALL FOR THE PARTS STORE, BRING IT IN AND WE CAN TEST IT. They cannot test starters. They are lying. All they do is put power to it, if it spins an pops out, they call it good. Fact is, it has to have a full load on it to test it. They cannot do that. So just buy a new one and try it. Should fix ya up. Your mileage is right on the common need a new starter mileage as well. Im on my 4th, now have the denso, probably wont need another.
2002 F350 Super Duty, Crew Cab,Long box,4x4, 7.3 PSD, Complete rebuild .010 over by myself except balancing, 2000hp billet connecting rods, ARP main and head studs, ceramic coated pistons, Smith Bros Chrome Molly push rods, Comp Cams springs, shims and seals. Competion complete valve set and head grinding and vacuum tested, Adrenaline HPOP, DP Tuner Infinity DX programmer and full load of tunes modified for my new Stage 3 250 shot injectors, ceramic coated exhaust manifolds and up pipes and collector, BD Diesel triple clutch furnace brazed billet torque converter and BD pro engine brake and compressor set up, MBRP 4" stainless turbo back, Driven Diesel regulated fuel return system, AFE pro stage 2 cool air intake, riffraff billet Whicked Wheel 2 compressor wheel in a GTP38R turbo, riffraff blue cooler boot kit and coil clamps, International Double Ceramic Coated Bellowed Up Pipes
1999 Mustang GT Convertible Anniversary, rare stock electric green
2009 40' Raptor Velocity Toy Hauler