Which protocol, as a part of TLS, handles negotiating and establishing a connection between two...

The TLS handshake protocol is what negotiates and establishes the TLS connection between two parties and enables a secure communications channel to then handle data transmissions. The TLS record protocol is the actual secure communications method for transmitting data; it's responsible for the encryption and authentication of packets throughout their transmission between the parties, and in some cases it also performs compression. Negotiation and binding are not protocols under TLS.