You're right that "two-way radio" sounds too technical and specific. That over-specificity is superfluous in your two examples for a single reason:
We know what type of radio the officer is using because of how they interact with it.
In Examples 1 and 2, the officer is doing something to the radio that is only possible if the radio is a two-way radio. Because they used it to 'call for backup' (Example 1) or 'report the scene' (Example 2), the radio cannot be a music radio. People don't call for backup on a music radio.
On another note, your concern would be more appropriate if the only thing we knew was that the officer was listening to "the radio."
"The officer heard someone's voice over the radio."
That's ambiguous. The "someone's voice" here could be that of a fellow officer (on a two-way radio), or it could be that of a disc jockey (on a music radio).
In this new example, using "two-way radio" would be appropriate because it would address the ambiguity.