I tried to give my take on this the other day but wasn’t successful. Before you can wire such a circuit you have to grasp the concept of how a 3 way switch works. Once you grasp that concept the circuit should explain itself. All the connect this to this and that and travelers etc won’t make the light in your head come on and that’s the first light that needs to light up.
Villa’s first youtube shows it pretty clearly.
A 3way switch isn’t an on and off switch it is a selector switch commonly called a single pole double throw. In schematic form it looks like a Y connection where power comes in the bottom of the Y and has the choice to go out one of the two tops of the Y. it never gets shut off just gets directed that’s why we use two of the switches together, connected at the tops of the Y. Think of a 3 way switch as an A/B selector switch. With two of them wired as such, A wired to A and B wired to B. When both A’s are on you have an electrical path or when both B’s are on you have a path. A&B or B&A you have no path. The wires between A,A and B,B are what they call travelers.
If you are at one end of the hall and the lights are off it means the switches are A,B or B,A and when you flip that switch it makes it A,A or B,B and that’s a complete circuit and the light comes on. When you get to the other end of the hallway you flip the other switch and it makes the A,B or B,A condition again and the light goes out.
Here is the symbol for a normal light switch a SPST single pole single throw and also a SPDT single pole double throw such as a 3 way switch.
Here is how they work together as a pair.
Hope this helps.