Most importantly, let's not mess up tense and aspect.
In grammar, the word tense refers to the time period in which the verb of a sentence places an action. For each grammatical tense, there are subcategories called aspects.
Aspect refers to the duration of an event within a particular tense. In other words, the aspect of a tense allows us to describe or understand how an event unfolds over time. English has four aspects: simple, progressive , perfect , and perfect progressive.
Consequently, it would be incorrect to say things such as: progressive tense, continuous tense, or continuing tense as they are all aspects.
As regards the use of the three terms, progressive, continuous and continuing, all of them are correct, but progressive, statistically speaking, outscores the other two.