Indeed, this makes some sense now. As you rightly conclude, during your time in the forces you will have paid NI, so when you claimed IB you would have been entitled to it (IB was a contributory benefit). When the ESA migration took place you ended up on ESA© because of your prior contributions. The contention in the Rightsnet posts that Speedfreek linked to is that you should also have been considered for ESA(IR) since yours was a conversion award - I agree with that point of view but it seems the DWP may need some...persuasion.
Anyhow, presumably they've sent you an ESA3 form so that they can consider you for an IR award. Since you live alone there's obviously no partner's income to consider. If you have no other income (pension, rent from property etc) and no savings over £16,000 then you might well be entitled. If you are awarded IR, the SDP should be considered automatically.