Telling my supervisor about my medical condition
Contact the disability center first.
You haven't listed a location, but most developed nations have laws prohibiting discrimination against individuals with disabilities, and mandating reasonable accommodations for their disabilities. The university disability office will likely have a better understanding of the university's policies and procedures as well as what reasonable accommodations you're entitled to than a random professor will (unless that's their research area or something). Additionally, helping you is literally their job, so you might as well take advantage of their services; it's not going to cost you or your lab/work group anything to do so, after all, and they should keep the contents of your communications with them private.