Well, I think that line derives from the fact that the fifth Doctor might expect to regenerate in any normal circumstance, but he isn't sure if spectrox poisoning is something he can recover from - but otherwise, he thinks he's got eight more regenerations left to him. BUT we, the viewer, can infer an extra layer of meaning in it based on our secret knowledge that there were eight Doctors before Hartnell, which indeed makes Davison the thirteenth, and so there's a whole extra meaning in why it "feels different this time".
It's brilliant. It all fits. We can have our cake and eat it.