Can you back probe the connector?

This is using a Cat Spoon (7x-1710) it is dished and follows the wire to where you can touch the pin inside.

chances are you don't have the fancy cat spoon. this is what i am talking about with paper clip like wire.

Now if your plug looks like this you are not going to be able to back probe.

I believe diesel crawler is talking about doing something like this.
Back probing will give you the best reading as the voltage drop could be in the plug.
If you can back probe,
One end of the meter goes to ground at the battery!, make sure you have good connection. back probe each wire and tell me what the readings are.
Next switch the ends of the meter, putting the end on the positive at the battery, tell me what readings are on all three wires.
before you are done I need battery voltage.
All readings are made with the solenoid engaged.
recap, when you are done you will have 7 readings.
3 connected to ground at the battery.
3 connect to positive at the battery.
battery voltage, all with solenoid engaged.
Now if you end up doing the last Picture. Which is the jumpers between the connectors, make sure you use an 18 awg wire. To small it will get hot and throw your readings off. In a nut shell you will make same readings as above. However, I would take the wire that you have going from the positive at the battery, once again making sure this wire is 18 awg at least. Disconnect the jumper going to the pull in winding, on the vehicle harness side and with the hold in powered up send 12 volts to the pull in. Based on what you have said above it should pull in as it sounds to me like you have a voltage drop on the pull inside of the coil wiring. Which is typical.