This myth keeps getting repeated. It hasn't been true since 1949, when British subjects in the UK became Citizens of the United Kingdom and Colonies.
In 1983, the status of CUKC was renamed to British citizen (for those CUKCs resident in or closely connected with the UK: the situation in the remaining colonies was more complicated). At the same time, the status of British subject was officially restricted to those few British subjects who didn't qualify for citizenship of the UK or of any other Commonwealth country in 1949, and who were formerly known as "British subjects without citizenship".
So we are officially and legally citizens, not subjects.
I think I've heard something similar -- that subjects have duties while citizens have rights.
But of course, citizens typically also have duties -- commonly, the duty to take up arms to defend the state -- and subjects can legitimately expect a reciprocity of obligations from the sovereign (e.g. the enforcement of the "King's Peace"), which sounds quite a bit like rights to me.
(All of which is a verbose and not very coherent way of saying that I agree with you.)
In 1983, the status of CUKC was renamed to British citizen (for those CUKCs resident in or closely connected with the UK: the situation in the remaining colonies was more complicated). At the same time, the status of British subject was officially restricted to those few British subjects who didn't qualify for citizenship of the UK or of any other Commonwealth country in 1949, and who were formerly known as "British subjects without citizenship".
So we are officially and legally citizens, not subjects.