Hey Derk, Thank you so much for putting in all the work that you have thus far on Celestial Woodcutting. The 
script bot works damn near perfectly for what I use it for but I'd like to report a few things:
Game Mode: OSRS
Bot settings: Woodcutting guild, Chop Magics, Bank Woodcutting Guild
Bot run duration: -
Bug description: As a few others had said already it does bank the axe, however this is easily worked around with bank fillers. One thing that I noticed is that on 3 accounts that have the axe equipped (Doesn't keep axe in inventory) It works flawlessly with hours on end, however on another 3 accounts If the axe is kept in inventory it will sit with the bank tab open forever.
Steps to reproduce: Simply just unequip your axe and run the 
script bot. It should do it the same for you as it does me.
Have you found a work-around (manual): As mentioned, you can prevent the 
script bot from banking your axe with bank fillers. I've been able to get the 
script bot to function again by manually closing the bank, but it doesn't start from there, I have to walk it next to the magic trees, from there it functions as normal. Once it goes back to bank however it's rinse and repeat.
If I could help you in anyway shoot me a reply, Thanks again for all your hard work!