65amp wiring/trouble shooting

It looks like a standard regulator - or at least the wiring is the same as standard.
The blue wire into the regulator is the sensing wire. It is joined with blue wire feeding the rotor.
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Measuring voltage at that connection on the alternator will reveal what voltage the regulator beleives the system is at.

Another location is at the ballast resistor. Same circuit and should be the same voltage.
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The voltage at the battery, and at the alternator output stud should always be nearly the same.
When the key is switched on the voltage in the 'run circuit' blue wires should also be the same.
If there is a difference, it will be worse when the battery is recharging, the lights are on, or other additional items are turned on.

With your new, fully charged battery, even if there is a problem in the wiring, it will be less severe.