I also say I am a Product Designer just for convenience because when someone hears ID, they automatically turn you into an engineer that builds plants and factories.
To me, a Product Designer solely designs a product to produce and sale. That is all they consider.
An Industrial Designer also designs a product, but there is so much more involved. They design the product but they also design the experience that the end user gets using it that wants them to continue.
I associate more of a marketing feel to an ID’er whereas I get a more engineering feel from a PD’er