Wow your example is a good one. Both are correct. We use the definite article e.g. 'the' when we are being specific and the indefinite article e.g. 'a' when we are talking in general. So in theory when we say we are watching the television this should be in reference to a specific T.V. The fact that in common speech, both of your phrases are interchangeable, exposes a difficulty in giving you a definite answer.