As I see it, it went down like this:
1753: Linné describes the genus Ostrea
1793: Thunberg describes an Oyster as gigas and puts it in that genus
-> Ostrea gigas (Thunberg, 1793)
That is the original description and should be the original name of the type specimen.
Over a hundred years later...
1897: Frederico Sacco forms a subgenus (sottogenere) Crassostrea in the Genus Ostrea (Linné, 1753). It is not considered a subgenus anymore, today it has the rank of a genus.
So at some point, Ostrea gigas was taken out of Ostrea (this genus still exists) and into the Genus Crassostrea (or maybe into the Subgenus Ostrea (Crassostrea)).
It says here that Crassostrea gigas is a homotypic synonym of Ostrea gigas, so that also backs up that it was "just" a matter of nomenclature.
I could not find the original source that describes the basis of changing the genus. The earliest source I found mentioning Crassostrea gigas is from 1952, Adresses delivered at the Convention of the National Shellfisheries Association (page 82) and there is a reference to findings of Schaefer 1937. So if you look for that you might be able to go back and pinpoint the origin of it at some point.
The entire new Magallana Genus from 2016 (Crassostrea gigas Thunberg, 1793 is its type taxon) is also an interesting read, it seemed to have evoked quite a spat amongst malacologists.
Maybe you can find something in the Biodiversity Heritage Library that helps with your question.