[redbird68]

My lift is 12' tall, so I designed the ceiling height at 12'-3".
I was fortunate to be designing the entire home, and this dimension affected the entire house regarding elevation of first and second floor.
If your garage is already built, I would assume you can discuss this with the lift mfr/installer.
They can probably adapt the lift to fit your 12' height.
If you have not already, I would select your lift mfr/installer and start discussion with their tech.

I used a company that had installed lifts where I worked, as well as many other commercial/residential locations. They were very helpful to ensure dimensions, elec power, foundation, etc were all ready to provide for a clean install.

Actually had them come by last year to inspect the lift and they showed me how to do it myself annually going forward. Lift was installed 10 yrs ago and still is in great shape.