How to identify wires?

There are wire tracer/signal injector tools available for this sort of task. Here is a Harbor Freight example.

These are conceptually not that different from Mex's suggestions, but can be a little easier to understand and use in practice and have the advantage of working via an inductive pickup so there's no need to pierce or strip the wire insulation. That's probably not a big advantage in your particular situation.

You can also deduce which wire is which by selectively shorting wires and measuring the continuity at the other end. It can take at least a few trips back and forth and some care in labeling and note-taking to be accurate.

I work in the telecom industry and we use a Toner device that has a tone generator that attaches to a pair of wires on one end, then a Wand is used to find the tone on the wire on down the line in a bundle. Works very well for wire ID. I would think you could find one at a Graybar (or other type tech warehouse store) in your area. Or online.

Yes.

Get a 12 volt battery. Jumper test leads, and a 12 volt test light.

Turn MASTER WHOLE HOUSE CIRCUIT BREAKER OFF. Then every individual breaker.

Verify verify verify the breaker panel is dead.

Connect battery leads to load end out on the patio. White is neutral common so make that your negative battery connection.

Connect 12 V + to black wire in question

Take test light to breaker panel.

Stab the breaker outlet screws.

Which breaker lights up?

Ain't cool to have multiple wires connected to a single breaker, but if that's the case, remove wires to isolate then re-test.

At the load end, feeding a receptacle can make things tough. False negative readings. A GFIC receptacle makes it impossible. Take the receptacle out of it's duplex and connect wires directly to screw terminals. Hint. Unplug things if you get strange readings during the test.

If you have to, unplug everything in the house when testing.