The battery terminal and the small coil terminal connected to it by the jumper will obviously have power. The other small terminal as well as one side of the starter button will also read a voltage due to the voltage being passed thru the relay coil. The terminal that should NOT have power is the large one going to the starter motor. Internally there are some large surfaces that are at battery voltage
Here are photos of the inside of the solenoid. The battery terminal side has a large metal sleeve that directly connects it to the large flat plate which holds the relay. The entire plate and both sides of the relay chassis thus becomes battery voltage. While the relay coil terminals are supported by the same plate they are kept isolated by flat fiber plates and insulating sleeves. A small contact plate and internal screw terminal connects the relay contact to the two solenoid coils. The motor side terminal has several fiber insulating washers and special shaped openings in the flat plates which keep the terminal from touching and shorting to the support structure. Electrical path for the solenoid is shown by the blue lines.
It is hard to say exactly what has happened but I suspect one of the insulating spacer plates or washers might have been improperly placed or perhaps slipped out of position when the solenoid was rebuilt. There are quite a few of the spacers that need to go in a very specific order and place to isolate the battery terminal and relay support from surrounding parts. It is also possible and maybe more likely the screw connecting the solenoid coils to that inner terminal is a tiny bit too long or the terminal was bent and something is touching the relay side plate behind it with some resistance. The entire relay body is at battery voltage and if something is barely touching with some resistance, enough voltage could be flowing thru the coil to heat it and be measured on the starter terminal but there would not be enough current available thru the coil to turn the motor.