Since you didn't provide a picture, or a very helpful description of what you're looking at. I'll try answering your question by explaining how the switch itself works, which will hopefully help you understand the problem better.
Single Pole Single Throw (SPST) Pull Chain Switch
The pull chain switch that controls the light(s), is a single pole single throw (SPST) switch. It has two positions ON (Closed), and OFF (Open). Drawn simply, it would look something like this.
Switch shown in ON (Closed) position.
When the switch is in the ON (Closed) position, current is allowed to flow through the switch, through the light(s), and back to the the source (via neutral).
When the chain is pulled and released, the internal contact rotates 90° (1/4 turn) into the OFF (Open) position.
When the switch is in this position, current is not allowed to flow through the switch, and the light is not lit.
This is why the pull chain switch that controls the light(s) only has two leads.
Single Pole Multiple Throw (SPnT) Pull Chain Switch
The pull chain switch that controls the fan, is a single pole multiple throw switch. It has multiple positions, which allows it to control the speed of the fan. Draw simply, it would look something like this.
Switch shown in OFF (Open) position.
When the chain is pulled and released on this switch, the internal contact rotates 45° (1/8 turn) to the next position.
Another pull, another turn.
Pull again, turn some more.
One final pull brings the switch 180° around, and again to the OFF (Open) position.
By manipulating the output of this switch, the fan is able to whirl around at various speeds depending on the switches position. The number of output leads, will depend on the switch. How those leads are connected to the fan motor, will depend on the fan manufacturer. This simply illustrates the basic principle of how the switch works.
As always electrical work can be dangerous, never be afraid to contact a qualified Electrician
I searched far and wide to find a solution for this for my own remodel. The orange connectors used by most of the lighting manufacturers is a standard IDEAL product - you can order them online in large quantity or on auction sites in smaller ones. You can retrofit the HALO cans to be compatible, or even use the connectors (as I did) to connect to GU10 sockets to enable use of LED spots with LED cans. The IDEAL model is Ideal 182 30-682.
Then you can just quick connect to your wires, plug together and you're good to go. Surprised these are so hard to find given how prevalent they are in the LED cans and retrofit bulbs.
Best Answer
You've got it. Just make sure that in the light fixtures you use all black in one connector, all white in the second and all bare wire in the third (from your hand-drawn wiring, it looks like that's how you plan to do it). That way, the lights are wired in parallel so that they both get the full mains voltage across them when you turn on the switch.
Also make sure that the switch is connected to the live (black in this case) wires in the two cables, which means that no electricity can reach the light fixture when the switch is off (which could save your life if you accidentally touch the live terminal in the fixture when changing a bulb).