To me, 'I had a tea this morning' suggests that I'm a connoisseur of teas ! It suggests that I had one particular kind of tea, out of the many hundreds of types of tea, blends, teas from a particular plantation etc, there are. The listener would expect this statement to be followed by some explanation e.g. ' I had a tea this morning..It was a very light Darjeeling, that I'd not tried before and it was excellent'.
'I had tea' suggests I had tea rather than some other beverage or that I don't normally have tea but did this morning. Which meaning it is depends on how the speaker says it, what they stress.
If all that was meant was the simple observation that I drank a cup of tea, I'd say 'I had a cup of tea' or '. I had .some tea'