What other bones are in the arm besides the humerus? If you're referring to the ulna and radius, those are the bones of the forearm.

It's also important to note medical diagnosis vs colloquial communication. When documenting the fracture (again, not indicated in this video), one would specify the bone, location, grade/degree, etc. A "broken arm", as you state (or an arm bent in half the wrong way" would be ridiculously abbiguous. Where is the break/bend? Which direction? There are many ways your shoulder and elbow aren't meant to bend. The key here is specificity, although with an important exception.

When speaking to the family or patient, one would speak to their understanding. To an adult, it may be appropriate to use medical terms more heavily. To a child or a family that may not understand these terms, "broken arm" may be sufficient for initial conveying of the incident. Perhaps to you, the term "boo boo" may be most effective in getting the point across.