Learn English – Why are decisions “made” instead of “done”

Firstly: exercise is a noun

Exercise noun -

1. bodily or mental exertion, especially for the sake of training or improvement of health: Walking is good exercise.
2. something done or performed as a means of practice or training: exercises for the piano.

Secondly

Exercise means doing something with the body.

Doing in this sense is a gerund. This means that doing is behaving as a verb within a noun phrase. So doing something with the body is effectively a noun that includes an action.

The sentence itself is acting as a definition for exercise, which is why the part after means is a noun.

To further clarify why using to to do is not natural, if you look at the accepted answer to the question you link to, then you can read this:

The infinitival is more associated with change, the gerund-participle with actuality.

Since your sentence is a definition, it is actuality, therefore we use the gerund doing not the infinitive to do.

Both dived and dove are standard as the past tense of dive.

Dived, historically the older form, is somewhat more common in edited writing, but dove occurs there so frequently that it also must be considered standard:

The rescuer dove into 20 feet of icy water.

Dove is an Americanism that probably developed by analogy with alternations like drive, drove and ride, rode.