Matthew Arnold's "Dover Beach" is a dramatic monologue because the poet is addressing a silent audience. The effect is of one person directly addressing another, while the reader listens in. For example, tradition has it that Arnold composed "Dover Beach" during his honeymoon, and that the silent audience is his bride. This differentiates the dramatic monologue from the soliloquy, in which the speaker only addresses himself. Hamlet, for instance, when brooding about suicide, does so before an audience, but really he is alone with his thoughts. Arnold writes, "Listen! you hear the grating roar," etc., and by this and other means implies that he is not alone, and is passionately unburdening himself to another party.
I would agree with you. The poem is certainly a monologue, but not really a dramatic monologue, since it lacks the element of a speaker distinct from the author. But this is all in the definition, since some people don't regard the distinct speaker as part of the definition of a dramatic monologue. Terms in literary criticism don't necessarily have the precise and agreed meaning that terms in biology or math do.