Resolution

Thanks to Jonathan Chiche and Johannes Huebschman for the answer to my question. First off, according to the online Encyclopedia of Mathematics, relatively compact means having compact closure (I had called that conditionally compact; neither term is very evocative). Now to denombrable a l'infini, first Johannes wrote that it meant that the one point compactification had a countable basis at the point at infinity. Then Jonathan pointed to a '57 paper of M. Zisman that actually defined it to mean \sigma-compact. In the context of locally compact spaces, the two definitions are easily seen to be equivalent! Since \sigma-compact seems to be widely used, I will go with that. And now let us break off this thread. Michael