Hard to know for sure yet, but the following scenario is possible…
Old battery died because either it was tired or your charging system was tired. I’m sure you know that the cold is hard on batteries, that’s because chemical reactions work slower in the cold. Cold is good for battery storage, but not battery use.
Then the new battery died because it didn’t have much charge to start with, because it just came from the store. You could put a charger on it and see how it does the next day.
Is interesting that the new batt powered your lights, but the starter didn’t even click. It’s possible that the starter is having a problem too, but the odds are against two simul problems. The starter needs a lot of current. The other gear less so. So it’s not impossible for there to be radio and lights, but no starter.
You might want to check the voltage coming out of your voltage regulator. I’m not sure what the spec is, but the 12V battery is really 13.8V, so it’s going to need 14-15V to charge.
Ed’s idea about checking the grounding is good too. It takes a lot of current to run the starter, so a connection that would work for a little current, can be insufficient for the higher current load.
Are you sure that the starter produced no noise at all? Not even the click of a solenoid moving? If the battery has it’s 13.8V or thereabouts, then you might try giving the starter a good whack. The starter’s solenoid might be a little balky in the cold. You should be able to hear the solenoid’s click, even if your battery is a bit low, or a ground is imperfect.