Final answer:
The receptor protein cleaved to release a transcription regulator when interacting with Delta is Notch, which is key in the Notch signaling pathway for cell differentiation.
Step-by-step explanation:
The name of the receptor protein that is cleaved to release a transcription regulator when it interacts with cell-surface proteins such as Delta is Notch. The Notch receptor undergoes proteolytic cleavage that releases the intracellular domain, which then moves into the nucleus to regulate transcription. This process is a central part of the Notch signaling pathway, which is crucial for cell differentiation and development in multicellular organisms.