You are in a branch of government and therefor part of the overall Civil Service.
Your pension is not paid by private industry, but out of the taxpayers contributions. In my eyes, that makes you part of the overall civil service.
Your eyes are just plain wrong then. The Civil Service and the Armed Forces are both segments of the UK Public Sector, along with the NHS, State Education, the Police, the Fire Service, Government (at all levels) and (still) heaps of Quangos.
And it's not just semantics; to refer to the Armed Forces as part of the Civil Service is ignorant on more than one level.