Ich sitze am Tisch
can be used for every table or desk. So it is also posible to use it when you are sitting at your office-desk while writing on your computer, or when sitting on a table-like workbench while repairing a radio. But of course, you can also use it to describe the situation where you sit at a table in a restaurant or at home having lunch or any other meal.
Ich bin zu Tisch
means that you are having lunch. The original meaning is that you sit on a table while heaving lunch, but when you buy a sandwich and eat it walking through the streets during your lunch-breack, your colleagues might also say that you are »zu Tisch«.
Ich sitze zu Tisch
is another possibility to say »Ich bin zu Tisch«. But since you explicitly use the verb »sitzen« (to sit) instead of »sein« (to be), it is hard to interpret to eat-a-sandwich-while-walking-scenario into »er sitzt zu Tisch«