What is the difference beteeen a solenoid's make current and inrush current?
A solenoid designed for AC supplies will have an inrush current that can be significantly greater than the steady state current it takes when it has activated. It all depends on how the solenoid is initially supplied voltage and the part of the AC cycle that the voltage is applied. If you apply AC voltage to the solenoid at the peak of the waveform then there is little or no inrush current whereas if the solenoid is connected to an AC source as it passes through zero volts then the current waveform in the solenoid can reach an excessive value that causes magnetic saturation of the solenoid core and make the problem of inrush even worse.

Here's a table of numbers taken from the internet that can give some insight into the level of the problem: -



And here's what AC inrush current can look like when the applied AC voltage occurs at a zero crossing and at full peak: -



The top half of the picture above shows the problematic inrush current taking time to settle down. Solenoids, transformers, inductors and AC motors can all produce this effect because when lightly loaded, they are basically an inductor and the inductor current is proportional to the integral of applied voltage.