Engine runs(via bump start) but starter motor will not turn it over

Hi All,
My First post! So help me out if i go wrong.
I am now on my 5th hijet which is a 2003 1.3 van which has a problem that i'm hoping you can help me with. All others were 993cc so starting the learning game again with the 1.3 engine.
So as it states above when i turn the key i get all the correct lights on dash i think, when stater motor engages the engine turns it untill there is compression and will not turn any more as if the battery is flat but the battery is good i think. so i be leave the stater motor is at fault. About a week or so ago i had smoke(grey)  come from under drivers seat when i try to start her. I found nothing burnt/broken after disconnecting bat. so after connecting the bat back up she started up fine. was this a sign that the stater motor is on its way out! Was the smoke from stater motor?  
Cheers Vince

I would have said that it sounds like typical dodgy battery territory... except for the smoke bit. Could it be a sticking alternator pulley and\or dud belt or similar causing a slipping belt, as this might explain a possible source for the smoke and charging issues? That said, I would expect there to be some squealing from the belt and a burnt rubber smell too so a may be off the mark here.

Thk for reply sputnik

We (my brother ex bm tech) has checked bat its good. Have had squealing noise but only when you would expect it. Changing thermostat today(another prob at same time) and fan belt of at the mo and ok Not sure if they are linked. gauge high and cold pipes but full of coolant and not fan switch on.
My take starter apart today to clean not sure if the solenoid can come apart
2.80ish from euro car parts hope this will work looks same as what we took out we did not see the which way round the sat was but now now which why to put new one in coolant is the nice pink color.
Sorry for any poor any grimmer just rushing around at the mo
may need to get starter motor from some were my van has 55k on the clock and i'm in shepperton near m3 jct 1
thk again

We found the fault!!!

Bad/No earth from engine to chassis.

This can affect your instruments readings, my temperature gauge was reading high about 25%. Which was a bit alarming to say the least.

Not sure if this affected the fuel gauge which has been playing up.
This has now been fixed buy cleaning up the connector @ the back of the fuel tank with the locating pin thing holding it in place on the chassis.

Still not 100% sure what made the smoke, but did find a thin earth cable at the back of the engine which had expired in a big way.

I changed the thermostat which didn't need doing, but at least it gave me a chance to flush the rad.

So lesson learnt……CHECK YOUR EARTHS

I do have a pic of the old earth cable but not sure how to upload it!!!