Axes should only strip wood, not carve it
Carving tool should only carve (as it does already)
Axes should be able to strip live wood aswell, after that live wood stop growing, and eventually die. It called "Girdling"
https://en.wikipedia.org/wiki/Girdling
Girdling would result in easier wood harvest process. You can go, and girdl multiple trees, and just wait for them to fall. (random tick, similar to growth tick. Not instant.) Maybe even better, tree will lose all it's leaves first, at random tick speed, and just then fall.