Is this statement of conservation of charge circular?

You're right that it's a bit circular as stated. The more rigorous way to state a conservation law is something like:

The rate of change of [quantity] in a bounded system is equal to minus the rate at which [quantity] leaves through boundaries of that system.

A "closed system" is then a system for which both of these rates are zero, i.e., the [quantity] is not moving through the boundary of the system.

The version you propose, "[quantity] can neither be created nor destroyed", is closer to this more rigorous statement. But the rigorous statement is a bit stronger than this. If a charge were to suddenly teleport across the room, without passing through the points in between, this would satisfy your version of the statement; but it would not satisfy the rigorous version of the conservation law above. Moreover, it's perfectly possible in particle physics for charges to be created or destroyed, so long as equal amounts of positive and negative charges are created or destroyed. Your version of the statement would seem to outlaw these events, but the rigorous version does not.

No, the definition is perfectly correct and not circular at all.

For example, consider the principle of "conservation of sound", which states that if no sound enters or exits a closed system, then the total amount of sound in that system is constant. That is false, because I can clap my hands. Sound is not conserved, even if you don't let any come in from outside, because you can make it. You can't do that with charge, so the statement is nontrivial.

You could imagine a theory of electromagnetism in which the charge of 'fundamental' particles in the theory (let's stick to protons) is allowed to change with time. In these types of theories, closed systems do not have the property that the net charge within them is constant. The principle of conservation of charge is a statement that says that no such theories can describe reality as we observe it.

Your last paragraph is correct in that the principle of conservation of charge does imply that the only way for the sum of charge in a system to change is via transfer of charge in or out of the system. This does not preclude processes in which, for example, we may have a neutral particle decay into a positively charged particle and a negatively charged particle (in practice, beta decay), so the statement "charge can neither be created nor destroyed" depends a bit on what exactly you mean by 'created' and 'destroyed'.