Let me give you a counter-point. I was hired by my current employer as a software engineer. I love writing software. Unfortunately, in our team, we didn't have anybody who was good at managing product development. Features where thrown in willy-nilly, design often not well thought through because designers weren't given enough time and developers could just go develop.

I was/am the worst Software Engineer in the group (these guys are seriously awesome), so they had me take over product management and only spend part of my time coding.

I'm not an 'architect', I don't design the technical systems. I design the product systems, and implement features along with the rest of the engineers, but my time is split between managing the product and doing the development.

I fell into the role because I like considering the user/marketing/etc as well as doing the software development.

I've been putting "Product Manager/Software Engineer" on my email signature because I didn't know that "Product Engineer" existed..

As I work for a technical organization, I think they'll appreciate the product engineer label more.