'I live in a town' means that you live in a medium-sized conurbation. In general terms, a town is bigger than a village and smaller than a city. It has most of the amenities that you would need for daily life, but perhaps not all of them. By saying this, you aren't specifying which particular town you live in.
If you say 'I live in town', you aren't using 'town' as an uncountable noun. 'Town' can't be uncountable. The phrase 'in town' is like 'at school' or 'at work' ( or in British English, 'in hospital' or 'in prison') - it's a set expression where we omit the article.
If you say 'in town', you are referring to the centre of a town or city - this is probably where most of the shops, businesses and entertainments are. If someone 'lives in town', it means that their home is in what we might call 'the heart of the town/city'. A young person might like to live 'in town', to be close to the nightlife and amenities, whereas a family might prefer to live in a quieter residential suburb.
My impression is that this meaning of 'in town' is a British usage that wouldn't work in US English. I'd be interested to know an American English speaker's take on this.