Gotcha, I didn't realize it was blazed. I went there last year and it kinda just looked like a water tunnel that people snuck through. Seems fine in the name tag I guess.
As for trail names, generally I prefer the CT way of doing it. Of course, many segments of larger trails have their own names, and segments may be part of multiple trails, so having it in the relation only is often the least messy/d.r.y. solution. Of course, the downside is that the default OSM layer doesn't render trail relation names, which is probably why it can be seen as acceptable to do it in other places. Though to me that's kinda borderline "Tagging for the renderer" (see wiki page).