Apply only the part of the name change that’s actually covered by US jurisdiction.

Strictly speaking, per the EO, this is what they should have done. The EO defines the area to be renamed as:

the U.S. Continental Shelf area bounded on the northeast, north, and northwest by the States of Texas, Louisiana, Mississippi, Alabama and Florida and extending to the seaward boundary with Mexico and Cuba in the area formerly named as the Gulf of Mexico

You can see exactly where that seaward boundary is on this map: …wikimedia.org/…/File:US_ECS_Regions_2023.png

The area described is less than half the whole Gulf of Mexico.