Was the Thai alphabet really derived from the Sinhala alphabet?

According to legend, King Ramkhamhaeng, The youngest son of King Si Inthratit, the first king of Thailand, modified the Sinhala (Sri Lankan) Script to create the Thai alphabet. The first written Thai language is inscribed on a stone known to Thais as the "Ramkhamhaeng stele". However, The Thai alphabet doesn't look like the Sinhala alphabet. Thai alphabet looks more like the Khmer (Cambodian) alphabet. Thai was once a part of the Khmer empire during King Ramkhamhaeng's time. Was the Thai alphabet derived from the Khmer or Sinhala alphabet?