Since these doorbells are made to work with mechanical chimes, I assume I need to add a resistor into the circuit.
But, what is the best way to hook these circuits into the Arduino pins so I can detect the button presses?
The problem is,Your doorbell camera/Button requires a certain load to operate. That load has to be continuous at all times. When the doorbell button is pressed, you are essentially bypassing that load and putting power to the doorbell. The camera takes one amount of current, The camera will operate in series with the doorbell because it takes less current. That current is not enough to trigger the doorbell. Two reasons why your schematic will not work. 1, the load required by the camera is greater than the load required by the power supply that you're using. 2, The power converter that you're using has a very wide input voltage range. It will provide an output voltage regardless of whether the button is pressed or not. You need a secondary load greater than the load for the camera, such as a doorbell. Only then will you get your Arduino to be able to detect when the button is pressed.