Here is garage opener I used in this project.

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Here is the front side.

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Back side.

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Here is the Photoshopped overlay image of the back/front side.
You can solder directly to the button, but I used nearby terminals.
IMPORTANT: Please make sure that POSITIVE and NEGATIVE wires soldered correctly. If you reverse polarity your mod will NOT WORK!
Striped wire (black/gray) is NEGATIVE/GROUND wire
I drilled small hole in the side of the remote casing and tied a knot so wire will not slide out.

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I used connectors, so if for some reason I would need to use remote in the other car I could just unplug it.
I did have plug laying around in my garage, but if you want to get exactly the same one you can buy them here
EL Wire Connector set
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Plug attached to the wires connected to the button bunk.

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Remote assembled, and Velcro attached.

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Wires tucked in and remote is hidden behind OBD port (easily accessible if needed).
Now I can control my garage door with button #3!