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I think of it like this, roughly:

"Full-stack engineer" describes a person's technical knowledge: comfortable doing front-end or server work, maybe comfortable designing whole things from scratch.

"Product engineer" describes a person's motivations: primarily interested in writing code as a means to making a product, or improving a product.

Lots of engineers are primarily motivated by writing high quality code, or very performant code, or prefer to write tools (libraries/frameworks) over writing products using those tools.