a-b, and c-d needs to be measured in DC. And you didn't provide measurements for when button is pressed and button not pressed.a-b = 0,3 volt´AC
c-d = 0,8 volt AC
The Modernizr operates on DC voltage. Your barrel jack power supply input gets converted to DC voltage for the rest of the board.
The green connector is an input signal from your bell/transformer going to the Modernizr's cpu via an optocoupler. The important bit is that the optocoupler acts as the bridge between the bell and the Modernizr. It acts like a lighthouse in a storm, flashing on a torch to the Modernizr's cpu when it senses enough power on the connector input. There is no direct electrical connection. Within the optocoupler, communicating a bell-is-pressed signal to the Modernizr's cpu is done by flashing on an LED, hence the lighthouse analogy.
So what's this got to do with your situation? As far as the optocoupler is concerned, when there is enough power (more accurately, enough current) on its input, it lights up. This is where I asked for the DC voltage at a-b when doorbell button was NOT pressed, and when it is pressed.
You want to have near zero current at a-b when doorbell is not pressed so that the optocoupler does not turn on, so that it doesn't communicate an ON signal to the Modernizr's cpu.
What this sounds like is that there was enough voltage/current going into the optocoupler that it was turning ON all the time even when doorbell button was not pressed. By adding the resistors, you have essentially lowered the current going into the optocoupler. The current is now low enough that it is not triggering ON, therefore it is OFF. If it works for you, then great; problem solved.But connecting it like my picture below worked. And it showed off, when not pressed and on when pressed. But is that a safe way to connect it? If I do a more rebust connection better with soldering? Its ac power.
Note that the same could be achieved by having the 2 resistors in series (back to back) instead of one either side of the connector terminals. Or a single resistor of similar total value (~6.6k). Also note that it depends on the wires connecting your doorbell to the Modernizr, since wires have a resistance, the longer your wire, the more resistance it effectively has, and that could change the situation once installed versus sitting there with short wires on your test bench.