- You should not ever use inductive load from voltage regulator shared with MCU.
- Relays are massive eaters, and the 4.6 volts were actually caused by arduinos NTC kicking in to protect arduinos supply, the relays on your picture are designed for power around 5DC so you should design your circuit in that regards. I am impressed by the fact that little regulator on pro miny even survived it, meaning its likely to be reasonably made board.
The Pro miny voltage regulator is nowhere near as powerful as thatone on Nano or UNO and only barely manages to run anything more then the MCU itself.
- Watch your current. I am not sure whats the max current of your buck converter. Amazon specifications usually do not concern themselves with petty things like reality. SO it might not be strong enough to run all 4 relays. Though Colour is matching more often then wrong
- Again relays are massive power eaters and the lower voltage, the higher current so you can often lower your consumption by using relays made for higher voltage.
- Next batteries don't store that much power, get influenced by external influences and their capacity further decreases as the life goes. you can save a lot of power by using transistors/mosfets where possible.
It takes considerable additional design steps to make project run on a battery and replacing relays for bigger transformers or mosfets is often part of it.
Set any of the relays on, in arduino code and measure its voltage against ground.
If its 5V then arduino is sending signal fine.
If it is not, then check the resistance between its and arduinos ground. if its just few ohms look for issues on the arduino.
if resistance between grounds is big or even unmeasurable, then connect the arduino ground with relay board ground.
If you can measure 3-5V (Dependant on a pro mini version) between arduinos output and the ground then you don't have enough power on the relay modules supply.
Extra note:Make sure the arduino is not powered from buck converter as volage regulator has minimum voltage diference it can drop and could not be powered via regulator from a supply of less then 6.something Volts
As stated in Jroberts answer once everithing is working, you might want to disconnect the suplly to arduino from 12 volt source to Vin; and connect your arduino to your buck converter set at 5V (4.95ish-5.05ish) to 5V pin adding 10uF capacitor between the 5V and GND on arduino.
The buck converters are more efficient in dropping voltage, then voltage regulators, so that would extend your battery life. (this isnt majourisue as compared to the arduinos consuption is tiny compared t the consumption of the relays