Also no specific suggestions from me on types or where to buy.
Linear actuators indeed have the problem of size. They have to go somewhere.
The rig in that video uses a single winch and two telescopic rails. That relies on the rails being pretty stiff, and the whole rig being pretty well balanced. Hoisting with two wires would make it a lot more stable I think.
When it comes to keeping two winches in sync, instead of using two motors use a single motor with two spools attached to it. That way you can be sure both sides wind at the exact same rate. The only possible issue then is how the cable winds, as cable layers change the diameter of the spool, and with it the uptake rate. For your lengths I think that's not an issue.
A locking mechanism is also a good thing: something that slides in place and keeps it up even when the power goes out and the winches can spin freely. Small solenoids could do this.