Yes, he booked a table on behalf of Jerry. This is what I want to say. So do we need to say something different if we have to say this?
Yes, you need to use "on behalf of Jerry" or "for Jerry".
The actual phone conversation would sound the same regardless of whether John was simply lying about his name or booking the table for his friend/colleague.
(Phone rings)
Restaurant owner: Hello. Planet Delicious.
John: Hi. I'd like to book a table please.
Owner: Of course. When for?
John: Saturday the third of June.
Owner: OK. What time?
John: 7.30.
Owner: Yes, that's fine. For how many people?
John: Six.
Owner: OK. That's all booked in. Can I take your name please?
John: Jerry.
Owner: Great. See you on the 3rd.
John: Thanks. Bye.