I’ve never seen anything other than a 323 on cross city services, but I’m not a regular visitor to the area
My understanding is that the platforms are set up to accept doubled up 323s (2x3x23m = 138m) but could only accept a single 350 (4x20m = 80m) as a double 350 (2x4x20m = 160m) is too long
A 323 offers more flexibility to cope with quieter/busier services (running as 3x23m = 69m if needed) whilst a single 350 is “neither nowt nor summat” (as they say around here)
Or, another way of looking at it is “a 323 is 3 23 metre carriages” - you can’t forget how many carriages long it is or how long each carriage is, since that’s its TOPS classification (sadly I can’t think of any other trains with such an appropriate number!)