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I have:

• one 220 V AC -> 10 V AC transformer

  

• one push-button switch (at the door)

• two bells for two different places of the house; they should ring together when the (single) door push-button switch is pressed. Note: I don't want to replace the bells because I like their sound.
  Photo of a single bell:

  

Question: Why do the 2 following solutions fail?

Initially, there were 2 transformers connected like this, and it worked (unfortunately, the 2nd transformer is now broken and unusable):

Edit:

• with "case B" above, we have, at the output of the transformer:

  • 10.3 V AC when no bell rings
  • 7.7 V if either one of the two bell rings
  • 6.0 V if both bells are connected. Then only one bell rings.

• close-up photo of the other bell (it seems that it is the same model) (you can click to zoom):

  

• when the bell is not powered, we have 15 Ohm between the two wires. When the coil armature touches the right metal part, the two wires don't seem neither shorted nor broken circuit: we still have 15 Ohm. NB: I have only been able to push the right metal by hand, and I have not been able to pull out / make the armature go out.

I have:

• one 220 V AC -> 10 V AC transformer

  

• one push-button switch (at the door)

• two bells for two different places of the house; they should ring together when the (single) door push-button switch is pressed. Note: I don't want to replace the bells because I like their sound.
  Photo of a single bell:

  

Question: Why do the 2 following solutions fail?

Initially, there were 2 transformers connected like this, and it worked (unfortunately, the 2nd transformer is now broken and unusable):

(Schematics thanks to @Transistor)

Edit:

• with "case B" above, we have, at the output of the transformer:

  • 10.3 V AC when no bell rings
  • 7.7 V if either one of the two bell rings
  • 6.0 V if both bells are connected. Then only one bell rings.

• close-up photo of the other bell (it seems that it is the same model) (you can click to zoom):

  

• when the bell is not powered, we have 15 Ohm between the two wires. When the coil armature touches the right metal part, the two wires don't seem neither shorted nor broken circuit: we still have 15 Ohm. NB: I have only been able to push the right metal by hand, and I have not been able to pull out / make the armature go out.

I have:

• one 220 V AC -> 10 V AC transformer

  

• one push-button switch (at the door)

• two bells for two different places of the house; they should ring together when the (single) door push-button switch is pressed. Note: I don't want to replace the bells because I like their sound.
  Photo of a single bell:

  

Question: Why do the 2 following solutions fail?

Initially, there were 2 transformers connected like this, and it worked (unfortunately, the 2nd transformer is now broken and unusable):

I have:

• one 220 V AC -> 10 V AC transformer

  

• one push-button switch (at the door)

• two bells for two different places of the house; they should ring together when the (single) door push-button switch is pressed. Note: I don't want to replace the bells because I like their sound.
  Photo of a single bell:

  

Question: Why do the 2 following solutions fail?

Initially, there were 2 transformers connected like this, and it worked (unfortunately, the 2nd transformer is now broken and unusable):

Edit:

• with "case B" above, we have, at the output of the transformer:

  • 10.3 V AC when no bell rings
  • 7.7 V if either one of the two bell rings
  • 6.0 V if both bells are connected. Then only one bell rings.

• close-up photo of the other bell (it seems that it is the same model) (you can click to zoom):

  

• when the bell is not powered, we have 15 Ohm between the two wires. When the coil armature touches the right metal part, the two wires don't seem neither shorted nor broken circuit: we still have 15 Ohm. NB: I have only been able to push the right metal by hand, and I have not been able to pull out / make the armature go out.

I have:

• one 220 V AC -> 10 V AC transformer

  

• one push button (at the door)

• two bells for two different places of the house; they should ring together when the (single) door button is pressed. Note: I don't want to replace the bells because I like their sound.
  Photo of a single bell:

  

Question: Why do the 2 following solutions fail?

Initially, there were 2 transformers connected like this, and it worked (unfortunately, the 2nd transformer is now broken and unusable):

I have:

• one 220 V AC -> 10 V AC transformer

  

• one push-button switch (at the door)

• two bells for two different places of the house; they should ring together when the (single) door push-button switch is pressed. Note: I don't want to replace the bells because I like their sound.
  Photo of a single bell:

  

Question: Why do the 2 following solutions fail?

Initially, there were 2 transformers connected like this, and it worked (unfortunately, the 2nd transformer is now broken and unusable):