In my mind, those three titles would do the same thing, as they both are responsible for the interface experience of a product. The only difference is the company that is hiring decided to name it something different. I suppose an Interaction Designer might be more focused on the look and feel of something vs the experience.

Your best bet would be to compare the job descriptions of various jobs with these titles, but if I was looking for a product design job, I would open up any UX Designer and Interaction Designer jobs as well.

As for your second question, again I think it depends on the organization. For my current company, designers interact with engineering (among others) throughout the design process, showing them their progress and getting feedback.

Once we are confident on a design, our front end engineers create a prototype for us to further test with customers and check for any issues in front end implementation.

Once all of that is complete, another front end engineer embeds with the engineering team and helps them get it hooked up to the back end.

People I know who are Product Designers design tangible items. Things you can purchase from a store. But for us UX Designers, we design how users interact with tangible and virtual products.

In my view, Product Designer is someone between PM and designer, just like FE developer is kind of like a person between designer and developer.

And in China, many companies are used to describe UI and UX in this way: UX is someone who is responsible for the interaction design; UI is someone who is responsible for the visual design. Although in fact, UX should include UI.

In my country there's a difference... a Product Designer is someone who designs a tangible product, selects the material, and makes simulations for resistance and stuff, perhaps a little coding if necessary.

UX/UI designer are the "same" they do the design of an app or website, but only the design and pretty much every UX designer doesn't know anything about User Experience, good practices nor trendings on web and apps.

Interaction Designer we don't have much of that kind of designer but are the ones that focuses, I think, in the merge of a product designer or hardware with the UX designer.