Thanks everyone for your help. I tried to attach this reply to my
original post, but for some reason it is not working.
I was able to solve my door bell button problem by researching online.
It ends up that I have a door bell that one can select different chimes.
In order for it to work, a small wire with a diode on it must be put on
the new door bell button, and wrapped around the two screws. I did
wonder why my new button did not have this wire on it.
It took me three minutes to fix the problem, and two hours of wasted
time trying to figure it out. Oh well, I learned something new, and
saved some money.
Thanks everyone for your help.