Learn English – Waving your shirt to cool yourself off

I would go with "dress":

"wear" being like "dressing oneself".

"feed" and "dress": properly reflexive verbs

The concept of a verb applied to oneself reminds me of reflexive verbs (a verb whose semantic agent and patient (typically represented syntactically by the subject and the direct object) are the same.)

More specifically, Properly reflexive verb, where the agent is simultaneously the patient. The verb is typically transitive, and can be used in non-reflexive meaning as well.

Peter bathes himself. Peter bathes the cat.

Here, "feed" or "dress" would be a properly reflexive verb.

"eat" and "wear": common ambitransitive verbs

An ambitransitive verb is a verb that can be used both as intransitive or as transitive without requiring a morphological change. That is, the same verb form may or may not require a direct object.

When the subject of the intransitive form of the verb is an agent (like the subject of the transitive form), so that the verb aligns the syntactic roles S and A, then the verb is a common ambitransitive with an optional object, and the intransitive version is an unergative verb.

So the question which remains is:

"What would you call the relationship between:

• properly reflexive verbs and
• common ambitransitive verbs

?"

^{I'm getting the feeling that the answer is waaay more simpler than this detailed grammatical analysis ;)}

There is a difference in the meanings; to be "in shirt sleeves" means that your shirt sleeves (long or short) are exposed - so you're not wearing a jacket, jumper or coat over the top (apart from a waistcoat of course). Like you, I've heard one or two people confuse the terms, but according to the Cambridge dictionary, among others, being in shirt sleeves does not imply short sleeves.