Think you need to make a few phone calls and check the rules in your area. When my DTs were eligable for the vouchers, we had such a hard job getting our nursery to admit that they HAVE to offer "free" places. My DTs were already doing three half day sessions, and I didn't actually want to change that, but had a friend who wanted her DD to do a couple of sessions a week (not up to 15 hours) and wanted to know if she could have that for "free". After much questioning and pushing, phone calls to the relevent people at the council and searching for info on various websites, the nursery finally admitted...
They are allowed to cap the number of hours per session that they will accept vouchers for (in ours, the session is 4.5 hours but they will only accept government funding for 3 hours) and that parents are then expected to pay for the extra hour and a half plus "extras" (lunch included in morning session, tea in afternoon session) BUT they HAVE to offer the "free" places (as long as parents can ask the right questions...!) which would mean a DC attending for three hours within the 4.5 hour session, and only during school term time. All snacks and/or meals would have to be provided by parents. AFAIK, legally, any setting which takes government funding must offer this, you just have to dig to get to it.
So your children's centre is giving you incorrect info. Of course, the other side of it is that if they're oversubscribed, they can probably get away with it - if a parent pushes, then they might simply discover that they have no places.
OP, if you want a "free" place, then your best bet would probably be a playgroup, which runs in term times only, and does shorter sessions.
Incidentally, after finally getting to the bottom of it with our nursery, my friend decided that she wanted her DD to do the full session anyway - that having lunch with the other children was part of the experience and so on. Still, it was nice to finally uncover the truth of the situation