Like I said before, these solenoid coils are pretty inexpensively made, most I have ever seen are noisy to some extent. The buzzing stops when you release the button because the coil is no longer energized when you release the button.
applying only 10 volts to a 16 volt chime probably doesn't help much, it will try to draw more current (that probably isn't available) to try to pull the plunger of the solenoid.
Exactly, with the new info then we can say the chime is surely undervoltaged, solenoids always require the most current at pull in, after they have pulled in the current drops, so with only applying 10 volts to a 16 volt chime your lucky if you even get it to pull in enough to have the plunger hit the top tone bar, and if it even does it wont stay there causing the plunger to just vibrate I bet your only getting one tone as it mostlikely is not hitting th efirst one but is hitting the second tone bar when it falls back down, put a 16 volt transformer on it and I would bet the problem goes away.
These chimes require the plunger to over travel to strike the tone bar then it falls back down back where the plunger tries to center in the coil until the button is released in which it falls down striking the second tone bar on a two note front door hookup a spring moves it back off the second bar so it can vibrate, if there is not enough voltage and the button is held a little longer then the plunger is not held in the center of the coil causing it to just loosely vibrate, the more closer the coil is to the rated voltage the more the plunger will be held in the middle of the coil where it is design to be held.
Also with the longest run at 15', or 30 feet the 24awg phone wire shouldn't be a problem depending on how far the transformer is from the chime, remember the chime circuit is just one long series circuit starting at the transformer through the button to the chime then back to the transformer, so the length of the run including the transformer will contribute to voltage drop.
Since these chimes operate on very low voltage to start with, voltage drop can be a big problem as 5 volts at 120 might not be a problem but 5 volts at 10 is a 50% voltage drop or half of your supply voltage, and voltage drop is based on current times ohms so it will be the same for a 120 volt circuit as it would for a 10 volt circuit so this is why it is very important to keep it very low when dealing with low voltage circuits, at 10 va we are talking about 1 amp at the most so this is why I said the phone wire shouldn't be a problem at 30' of wire but any longer then you could start seeing problems.