"Self" bow, "traditional" bow, and "compound" bow are all adjectives to explain 'what kind of "bow" (noun). "Crossbow" is a noun of it's own and depicts something different than a 'bow"
While it likely seems splitting hairs, one might be not like eating "groundhog" or intend to saddle up a "seahorse" at the cattle roundup. Usually words of this type mean something totally different than when they are seperated as your first examples did.
While I do understand choice is a wonderful thing, one truly has to understand that 'choice' isn't the sole domain of a crossbow user in this argument. Any season based on equipment restrictions will have those whose weapon preferences lies just outside the lines. They also want "inclusion" and can use a similar debate style.
In the end, my own belief is that any open season is fair as long as it is open to all to make the ultimate free will choice to participate or not with the weapon structure allowed by the state.