Not that I'm sure I agree - how many other stations match these clues and have either an island or terminus platform? - but whatever.
There is a second station in this town, which functions as a terminus at peak times. There are only five letters in the town's name.
If anything, I would suggest I've been overly lenient with the clues so far; if you don't know where to start, just post any old station that is at least remotely feasible (i.e. a non-electrified station outside of England in this case, as Matt did).
EDIT: Sorry; I see you've suggested Cumbernauld. As you can surely tell from the above, it is not correct.
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OK, I'll let this go so we can forget it. I think I may have a second migraine coming on in one day so I don't want to be on here much longer.
OxtedL, this is yours to lose. It begins with "L" and is in Northern Ireland.