Sorry, still not clear what do you mean by "not touching"?
Some things that can be done (not exclusive):
You can use a remote with setting "Rec key" to trigger recording.
And you can use Lua, of course.
I just triggered my 650D by voice after activating "Audio Remote Shot". It will take additional pics, though. But movie start/stop is not a problem.
More details, please!
I need this to shoot a time-lapse. The camera must not move even a millimeter - therefore, you must not touch it. Sometimes during breaks you need to switch to video shooting. Sometimes, during breaks, you need to switch to video shooting, but do not move the camera. If I manually switch the mode on the video - using the mode wheels, then there is a possibility that the camera will move.
Therefore, I am looking for the simplest and cheapest solution, how to switch from time-lapse recording mode to video recording mode and vice versa.
I found an application for android "DSLR Controller", judging by the photo of the application interface - you can start recording video, but I'm not sure if you can start recording without changing the mode from manual to video recording. I can't test it now, because there is no mini usb -> type-c cable. I ordered a cable, I'm waiting for it to come and check it out.