TV or television has several meanings that are relevant to this question:
- (uncountable) An electronic communication medium that allows the transmission of real-time visual images, and often sound.
- (countable) A device for receiving television signals and displaying them in visual form.
- (uncountable) Collectively, the programs broadcast via the medium of television.
(All quoted from Wiktionary.)
It follows from this that "watching TV / television" does not mean the same thing as "watching a / the TV / television".
- Watching TV means watching programs that are broadcast on television. So when someone says, "I watched TV at home", they mean they watched (part of) one or more programs that were broadcast on TV (through whatever broadcasting technology, such as cable, internet or satellite).
- "Watching a TV"–if that phrase would be used at all, since it sounds unusual–means looking at or observing a television set, e.g. to see whether it is working correctly. This activity isn't nearly as common as watching TV; we usually just turn on the device to see what's on. Unless you are an electrician or interested in hobby electronics, you will rarely watch "a TV" or "the TV". (The definite pronoun would be used only if it is clear from the context which television set you have in mind.)
The following quote from the University of Warwick's website contains several examples of "TV" (without an article), "the TV" and "a TV" (my emphasis):
It is your responsibility to ensure the TV you are using is licensed. You could be heavily fined if you are caught watching a TV without a licence. (...) If you watch TV on a laptop the same rules apply, (...)
In this case, if you have a TV in your room, you will need your own TV Licence. However, if there is only one TV in a communal area, then only one TV Licence is required.
If you don't watch TV in your house (or room on campus) at all, or only watch TV programmes on a computer after they have been broadcast then you should not pay for a TV licence.
In the above example, the phrases "the TV you are using" and "watching a TV without a licence" are about devices for which a licence is needed, hence the articles.
Below are a few example of "[verb] at the TV" or "[verb] at a TV" that show in what ways these phrases can be used (my emphasis):
- "A decision went south, relationship went wrong, a life went sideways, and there they are, staring at the TV, watching through their pain while others are enjoying themselves." (Dave's News, 25.12.2018) The subject of this sentence looks like they are watching TV but they are probably not taking in the content; hence, "staring at" and the definite article.
- "I watched his speech from a bar in downtown San Francisco, as fellow patrons stared at the TV screens, stunned." (The Independent, January 2017) In this case, people are not staring at the tv but at the TV screens that were in a specific bar in San Francisco.
- "It is quite difficult to access where the fault is coming from till the engineer diagnose the fault. Many times customer do ask us that surely we would have come across this fault before, to which we reply that yes we do and we do try to give a round figure as to how much it may cost them, but because we have not diagnosed the fault or have looked at the TV we can only guess." (Hitachi TV repair, no date) This example is about looking at the device, i.e. inspecting it, to find out what's wrong with it.
- 'Mr Fenton said he left the control room to look at a TV and saw the side of Grenfell Tower was "fully engulfed in fire".' (BBC News, 17 July 2018) In this sentence, "a TV" means any TV screen where one could see what was going on at Grenfell Tower.
Sentences with "stare(d)/staring at a tv" are hard to find; you typically find the phrase "stared/staring at a tv screen", because "TV screen" is by always a countable noun.