If I had to name a single date I would say
August 14, 1963.
Seriously though, I can't find an exact date and there almost certainly isn't a single date. It sounds like in some groups they are still simply referred to as "dolphin". (
1)
The guess that makes the most sense to me is that it started changing with the change in status of whales and dolphins. Prior to the 1960s (+/-) whaling and catching dolphins didn't have the taboo it now has; whaling was still legal until 1986. Dolphins (mammal) were often caught with other commercial fish and either processed or wasted.
When humanity started paying more attention to and caring about our relatives in the seas (see link above) it probably became more important to distinguish between "dolphin" the fish and "dolphin" the mammal. Especially for commercial fishers and restaurateurs who don't want to offend people by letting them think they were catching and serving Flipper for dinner.