You don't really need much by way of accessories to signal that you're running for exercise, just some clothes that are clearly more appropriate for accommodating sweat than for fashion, and probably a water bottle if you're running long distances.
That is also a valid point. I don't currently have any exercise clothes beyond old shirts though, so I need to buy some anyway. My wardrobe primarily consists of 'fade into the background' gear, and school uniforms, with an old suit jacket kicking around somewhere.
An exercise:
Name something that you do not do but should/wish you did/are told you ought, or that you do less than is normally recommended. (For instance, "exercise" or "eat vegetables".)
Make an exhaustive list of your sufficient conditions for avoiding this thing. (If you suspect that your list may be non-exhaustive, mention that in your comment.)
Precommit that: If someone comes up with a way to do the thing which doesn't have any of your listed problems, you will at least try it. It counts if you come up with this response yourself upon making your list.
(Based on: Is That Your True Rejection?)
Edit to add: Kindly stick to the spirit of the exercise; if you have no advice in line with the exercise, this is not the place to offer it. Do not drift into confrontational or abusive demands that people adjust their restrictions to suit your cached suggestion, and do not offer unsolicited other-optimizing.
To alleviate crowding, Armok_GoB has created a second thread for this challenge.