After having asked on the official [D&D facebook][1] page, i've gotten the answer that it's not possible for a Wizard to learn a spell from a magic item the way i've described it. Here's the direct quotes from the person(s) i was in contact with.
The rules on page 114 of the Player's Handbook are pretty specific as far as to what you can use to copy down a spell. Generally, the rules do not support learning spells from magic items that perform similar roles.
You are, however, more than welcome to work with your Dungeon Master on a house rule for this if you feel like it should function differently!
The rules do only support written sources, as you indicated from that section. Something like a spellbook or a scroll would be able to provide the verbal incantations necessary for a spell, while an item may not necessarily do that.
These are the means that are explicitly provided by the rules. Again, you are more than welcome to coordinate with your DM about a house rule!
We see that the major reason why my theory falls apart (sadly) was due to the wannd not being able to provide the "verbal incantations necessary".
However, as they wrote on their quotes, they encourage house rules, so while this isn't book legal, if your group and DM decides to use it, then you're free to do so.