Is what the worshipper did correct: bowing, thinking the Imam had bowed, instead of performing the prostration of recitation, then realizing his mistake and prostrating with the Imam, then performing the prostration of forgetfulness after the taslim?

Your prostration when you realized your mistake was correct. However, the prostration of forgetfulness (sujud al-sahw) after the prayer was not prescribed, because the Imam bears the forgetfulness on behalf of the congregant according to the majority of scholars, and this has been narrated as a consensus.