Based on your history and the photo, this does not look like an
HIV related problem.
You had vaginal sex using a condom, and when the condom broke you immediately changed to a new one. This significantly reduces HIV risk. HIV is not transmitted just by brief exposure, and it does not cause local burning or skin changes on the penis head within 2 to 3 weeks. HIV symptoms, when they occur, are usually general symptoms like fever, sore throat, body aches, lymph node swelling, or rash over the body, not isolated burning at the tip of the penis.
The burning sensation and mild irritation you are feeling are much more consistent with local causes such as friction during intercourse, mild balanitis, sensitivity to soap or oil, over-cleaning, anxiety-related hypersensitivity, or a mild fungal or irritant dermatitis. The absence of discharge, ulcers, blisters, or pus is reassuring. Applying coconut oil repeatedly can sometimes worsen moisture and irritation, especially if the area stays damp.
This is not an HIV symptom. From the exposure described, the HIV risk is extremely low to negligible. If anxiety is high, an HIV test at 4 to 6 weeks (4th generation test) can be done purely for reassurance, but medically there is no indication that this is HIV.
Next Steps
Advice for now is to keep the area clean and dry, avoid oils or harsh soaps, wash only with plain water, wear loose cotton underwear, and avoid sexual activity until the irritation settles. If burning persists a mild antifungal or soothing topical cream will be useful.
Health Tips
please try to control the anxiety.
HIV fear itself can cause heightened genital sensations. Based on your description and image, there is no sign suggestive of HIV.
Good luck!