Michael, that panel looks pretty much the same as the panel for my HS-era layout, except I was earning my Radio Shack money as a grocery bagger and didn't use three buttons for a crossover where two would do. Your circuit diagram is just fine, except that my experience is that most (?all?) twin-coil machines work on AC of the right voltage/current, but buzz loudly. Using a DC supply of similar rating (or just buying a rectifier bridge when you buy the pushbuttons) will be much quieter.
I drew out layout plan as you confessed that you struggle with wiring. The diagram first posted is good but hard to figure out for you particular question. I drew a plan that is what it would look like on your control panel.
When you take the wires to the switch machine you have a 50 / 50 chance of driving it the correct direction the first time. Just switch solenoids and they will cooperate.
The plan here shows that one button can do two things. The crossover button will drive both turnouts. The "through" buttons are redundant so that either one will set both tracks back to straight through. It makes things very easy for visiting operators.
Let me know if you need additional explanation. For clarity I showed only the power circuit and drew ground symbols that will share one side of each solenoid. It doesn't matter which side of the solenoid is ground. It's just making a momentary magnet.
Don't worry about it - Slim (Fire is a Good Thing) Jerkins will jump in here after you hook it up and tell you how 'It Shouda' Been Done'!! If there ain't no sparks, you didn't work hard enough at it!!