Citroen Xsara Picasso cranks over slowly battery is fine

faulty starter motor probably

i would try a new starter moter sounds very much like this is your problem

This could be the battery - terminal to chassis cable connection. The starter motor is a power greedy item especially in cold weather. Any voltage drop through a bad connection anywhere in the circuit will inhibit operation. Are you really sure that the battery is in good order? Trying it in another vehicle of a similar size is one way. If you are confident with a multimeter set it on a low voltage DC range (9 -20V) and measure across all connections whilst an assistant cranks the engine and see if you get a significant reading, say 3v+. A battery that seems in order can also fool you, connect your meter across the battery terminals and crank the engine. anything under 9v is suspect. Check cleanliness of ALL connections before parting with cash.There are more tests you can do, such as checking the cranking amps but you need a clamp meter that measures dc for that.