Mayan Falls was the place we were looking for. It is run by a vacation rental company. The rental had 3 floors and a total of 9 units. The first two floors are single bedroom units, and the third floor is a suite with a few rooms. The pool is lovely. All the units were named. You actually get the exact room listed on the booking website. Mayan Falls looks as described in the listing pictures. There were several bikes available on a first-come, first served basis. Fortunately, there weren't that many occupants during our time there, so we had free bikes during our entire stay. It only took us 10-12 minutes to leisurely bike to where the restaurants, bars and ferry docks were located.
A few weeks before our vacation, the concierge (Veronique) who lives in Belize city reached out to us to make sure they are prepared for our airport pick-up and drop-off, tours, restaurants and any information we needed before our trip. She was very kind, made sure we were picked up and dropped off at specific places, and also that we had everything we needed. Veronique even had her husband picked us up from the ferry to the airport and got me Paracetamol for my pain. The manager of Mayan Falls (Chantel) on Caye Caulker always promptly responded to my WhatsApp messages and phone calls if we needed a taxi pick-up or anything. Overall, we had a great experience as the staff and their assistants work together to make sure we had a wonderful time during our stay.