I track my skinning and xc skiing in TP, I classify them as XC-Ski activities. I mark lift-served skiing as Other so I can delete the vert and omit it from certain reports, otherwise it looks like I climbed a lot more vert than I did. If I want Pa:HR metrics I just temporarily switch to Run, then note the numbers in the comments field and switch back to XC-ski.
I don’t climb so I can’t make a recommendation for how to categorize it but I’m sure someone here does! It seems like the Custom field could be used for that.