Of course I mean the element oxygen. If I meant the molecule, I would use the term "oxygen gas" or "molecular oxygen".
Problem is when people are referring to "living without oxygen", they mean living in anaerobic conditions completely. The fact that oxygen is contained in other molecules is irrelevant, since nobody is implying anything about elementary oxygen (lots of other elements are required for life too, like C, H, N, S, P, and a bunch of metals). But nobody is saying "we can live without carbon". All organic molecules are made of carbon, but when we talk about carbon in trivial usage, nobody means elementary carbon. They mean C(s), ergo, carbon solely. (of course with carbon the problem arises that you have several stable forms like graphite, diamond, etc).
Elementary oxygen is bound to so many other things (carbon, hydrogen, nitrogen, phosphorus, sulphur) that it is useless to refer to oxygen in the form of the elementary substance. While you are technically correct it doesn't serve any purpose other than confusing people. When researchers are looking for oxygen, they are looking for the molecular form in which it is common on earth. Besides that, you are rarely going to find O on its own. Oxygen would be a radical in that form and react with pretty much everything.
Even molecular oxygen reacts with almost anything given the right circumstances. Almost every metal that exists, with the exception of a few things like gold, is directly oxidized in air, and the process speeds up in water. If you find elementary oxygen, it could just be ferrous oxide or something, or silicon oxide, or aluminium oxide, or whatever. All that means is that once upon a time, in the beginning of the universe, there may have been a bunch of oxygen on a planet that immediately reacted away with the minerals.
Bottom line: if you say "life can't exist without oxygen", you're correct, but you're being unnecessarily trivial and odd about conventions. It's much, much more convenient to define "oxygen" semantically as molecular oxygen because you would always find oxygen around as a diatomic molecule. Solely "O" atoms don't exist. If people are looking for oxygen they are looking for oxygen compounds.