Now the outside unit doesn't turn on. Fan or compressor. Well, kind of.

I went outside, took a look at the control access panel. There's a plastic switch type thing that if I press in, the unit turns on. Both the fan and the compressor come on. However if I let go then it stops running. Using the thermostat control to turn the unit it doesn't work.

I assume the unit overheated and shut itself off but I can't figure out how to reset it back to normal.

The blower inside downstairs works fine.

Any Ideas before I call my local HVAC technician?

Sounds like you`re not getting low voltage 24-27 volts coming to the contactor telling it to pull in and kick unit on
Look over low voltage tstat wires from outside condenser where it hooks to the contactor trace all the way to the furnace, look for chews, cuts, bites,breaks,, after this get out your multimeter and atart looking for where the lowvoltage starts and where you loose it

As for the cover, I`ve never believed or understood why people use them. They just cause trouble just like what you almost did by forgetting to remove them. An old trash can lid on top and a cement block to hold it down is all you need to keep the fall leaves out.

Whooooa,

Just wait awhile. The unit probably has a hi pressure switch that kicked it off (did its job) when you left the cover on. No big deal just wait until the pressure equalizes (hour or two max)

You pressed the contactor in (plunger) and got it on and that's a good test.

On newer units there is a high-pressure cutout safety switch inside the outdoor unit with a red reset button you need to reset. Can be hard to find and require covers etc to be taken off to get at. I would get a tech to do an annual service.