Does the Apple Watch belong in a conversation about time pieces? 

Depends on context. Unfortunately for people that are into "horology", I would say no, but not because it's not a time piece but because when people generally talk about horology they are appreciating the history and craft that goes into the watch or the brand.

Unfortunately the modern smart watch is just too new to have the legacy and nostalgic  sentimentality that Casio elicits despite being mass produced, and it definitely does not have much artistry put into its production beyond design.

Perhaps with enough time people will look back with a new sense of appreciation on iFixIt's gadget tear-downs and Scan of the Month posts that show the care and attention that go into circuit design and chip placement of various devices.

Or do you think it doesn’t belong here at all?

I think there is a place for it, albeit small. If someone wants to be able to set multiple count-down timers, while running a stop watch, while keeping many alarms, it's the right tool for the job. A mechanical watch simply cannot do that, or if it can the complexity required translate to a cost that is prohibitive to most of us.

Do you like the Apple Watch, or do you hate it?

I like it. As other commenters have said, it is the modern tool watch for the office worker. 

I enjoy mechanical watches because they are simple and help me track time without tracking that much time. I like the idea of the watch lasting years without needing a service and that it can be powered by my winding instead of needing the full infrastructure provided by modern civilisation.

A true enjoyer of timelessness and mindful appreciation of life would probably not even want a watch at all.

The apple watch is on the complete opposite end of that mindset. I can multi-task to my heart's desire. I can have as many notifications as I want. I can track and quantify various things about my life beyond just how much time has elapsed since I turned my watch bezel.

Follow up question - do you wear an Apple Watch on the other wrist?

Just one watch at a time for me.

Or do you rotate on different days between the Apple Watch and others in your collection? 

I daily it on work days and change to a mechanical on weekend, when it's useful to know the time, but I don't need to be that on time for anything.