What is the correct way to wire a 3 way switch where power comes into the middle switch?
You'll lose your mind dealing with all the red black white business.  Get a 5-pack of colored electrical tape and start marking wires for their actual function.  Always mark wires the same at both ends.  Here is what's really happening in a 3-way.

First, the two traveler wires get marked yellow.  There is no need to tell them from each other.    So in the /3 cable, tape the black and white wires with yellow tape.  That'll leave red, which is a preferential color for switched-hot.
On the black and white cable up to the lamp, mark both ends of the black wire with red tape, since it is switched hot.  Mark the lamp's black lead red also.

The hard part is behind us. Now we wrap up.

On both 3-way switches, the yellow wires go on the brass screws.
On the near 3-way, the supply wire (black) goes onto the black common screw.  (It's almost like someone color coded it, lol).
On the far 3-way, you have no choice: 1 wire 1 screw.  Done.
All the wires that remain are white and red.  Nut them by color: white to white, red to red.
Golly, that was easy!  Or rather, we made it easy with our thinking cap and some colored tape.