It's funny how no one give you an aswer to the issue.
Go to your main disk - probably C/Users/AppData/Local/Frontier Developments/Elite Dangerous/Options/Bindings
Then you need to figure out which one of files is the one you are using. For me it's
Custom.4.0.binds
For you it's likely to be the same.
4.0 - Is Odyssey
3.0 - Is Horizons
In order to move bindings from Odyssey into base game, you need to copy past 4.0 file starting from
<KeyboardLayout>en-US</KeyboardLayout> and finish it with </SAAThirdPersonFovInButton> part.
You need to copy past everything that starts from this <KeyboardLayout>en-US</KeyboardLayout> (including it) and everything between it and </SAAThirdPersonFovInButton>.
I suggest to use Notepad++ for this.
Everything else after </SAAThirdPersonFovInButton> is related to Odyssey. Once you figure out how to do this, you can also copy past your binding from horizons to odyssey.
Thanks mate!
I still am struggling with a similar problem so you might be able to kindly help me with that

1. I did all the bindings in Odyssey, all was working fine.
2. I renamed the binds file to MUTZEL_WARTHOG_2.binds, changed the 2nd line accordingly and edited startpreset.start
3. Started Horizons, now:
- ED created a new MUTZEL_WARTHOG_2.4.0.binds
- ED changed the entry in startpreset to the standard ThrustMasterHOTASWarthog
I do not find your </SAAThirdPersonFovInButton> entry in the 4.0 file.
Edit: One of the weirdest things is that it created a 4.0 file when I started Horizons where 4.0 is supposed to be the Odyssey binds? °-°
Edit 2:
So I tested it again since I found that only three entries in my keybinds and the game created were existing (so the game created Horizons binds file included the "Humanoid" entries!
1. After copying my Odyssey keybinds over to the game created 4.0 binds file (incl. editing the 2nd line and changing startpreset.start) I started Odyssey. All fine. The game created four entries with my keybind names in startpreset.start (all correct):
MUTZEL_WARTHOG_2
MUTZEL_WARTHOG_2
MUTZEL_WARTHOG_2
MUTZEL_WARTHOG_2
2. I started Horizons, game wanted to start with generic keybinds. In the Options I switched to MUTZEL_WARTHOG_2. Then the game changed startpreset.start to only one entry of MUTZEL_WARTHOG_2.
3. Started Odyssey again and the four entries are back in startpreset.start.
At least now I have the same binds in both.(edited)