first i don't know much about cars,

i live in Kansas city, mo and right now it's like 9 degree here, i started my car it started fine after little nagging but my engine (which i believe is my engine) made kind of grinding noise (one like some kind of fan is grinding with its wall or something)
noise was gone right away and after driving to my friend's house, when i started my car at his house engine didn't make that grinding noise at all.

that was specific incident.
but everytime it gets kind of cold here and i try to start my engine, it makes that noise and once car is fully warmed, and try to start it i doesn't make that noise.

so what i was wondering should put dry gas and see it resolves the problem or it something out of my knowledge.

sorry for the big write-up, just wanted to have clear picture.
i have 2004 jaguar x-type 3.0 V6 manual transmission with 104K miles on it

 

i forgot to mention imp thing, grinding noise comes after the car has just started.
the moment car starts it makes that noise for halflsecond and then it goes away

 

very hard to guess without hearing the sound. The description makes it sound like maybe the starter motor pinion gear is a little lazy retracting.

I say this because your car is a manual with over 100k miles on it. The clutch dust is known to accumulate in the starter pinion gear, and cause it to be sticky. SOmetimes it grinds when you engage the starter, but from what you are describing it happens when the pinion should be retracting.

I would suggest that you find a helper and the next time it is cold. OPen the motor hood and listen carefully for the source of the noise.