OK, so how about we look at what you are claiming as part of you identity.

You are claiming to have done something. Which you have done.

You Hiked the Appalachian Trail.

A "hiker" is someone who currently hikes. An AT Hiker then would be someone hiking the Trail.

What you are claiming as part of your identity is "I hiked the AT".

Hence why when someone goes missing on the trail, they are identified as a trail hiker. If you went missing, you would be whatever you are, but I doubt that one of the identities they use would be "AT Hiker" because that term implies someone currently hiking the trail.

This entire time I think you are looking for the term "AT Conqueror".

But regardless, you're identity is as someone who has hiked the trail. But as I said what you do in your own mind is your own business. But it is not part of your identity as far as others are concerned because that is not something other people identify you as.

Let us look at a situation.

Person 1 says "I am a white catholic woman who hiked the Appalachian Trail"

Person 2 goes "You are a white catholic woman, hiking the trail holds no relevancy to me therefore I omit it from your identity as I see it".

If no one ever acknowledges you do something, then it is only part of your identity insofar as you claim it to be something that identifies you. Which can be your internal identity, I guess. It just seems strange to identify with such a singular achievement.

But your identity as far as society goes, is what other people identify about you. If they never identify that particular trait about you, then it isn't part of your identity.