Product designer makes sense in the context it’s used when you look at what the product is.
Spotify will hire product designers, even though their “product” is software. In the UI/UX world Product Design typically has it’s own connotation as a more generalist UI design role.
I’ve always found in the Product Design term was more common in Europe where a lot of courses were under the major of Product Design/Product Development compared to Industrial Design.
Either way, no reason you can’t do multiple searches to try and highlight what you’re looking for. The job requirements should make it clear what they are looking for whether it is a UX or ID role.