Burning Damage and the Elements
It's worth noting that the primary damage that comes from Freezing, setting fire to, and arcing electricity to, (and yes, also acid!) is to burn the individual.
In magic settings using the method of burning the individual that they are least resistant to makes sense.
Similar to "summoned" weapons and "summoned" armor Ice could also make for make-shift melee weapons. While it won't stand up to Steel very well, an ice blade can make for a quick advantage in an unarmed fight or a knife fight. Similarly, it could make for a mediocre shield. If it's really hard ice it may even stand up to a couple of blows.
The best of these would be an Ice-Hammer built out of a wood/iron shaft, because even if the ice can't survive due to it's durability, the weight of the ice on the shaft will still make a very effective mace for blunt-force-trauma; except you don't have to carry a heavy mace/hammer all the time, only the shaft.
As a tool
Finally, cold happens to be very useful as a tool for preserving food. You use it every day extensively via your refrigerator and freezer. Given this, old civilizations could more easily stockpile food for various occasions and probably increase their health by having better access to a variety of foods that won't spoil.
It can be used in a few other "defensive" applications as well:
blocking a bridge/choke-point with an ice wall
Patching a ship that has a hole in it. Not only will the ice plug the hole, but also has the property of being less-dense than water (like the wood near it); and the water to be used to patch the ship is coming through the hole right now! How convenient
People using Siege Towers or Ladders can simply be blocked by freezing the siege tower shut or walling off the position the ladder was place on
Freeze the mechanism of a ballista or catapult to keep it from firing (though this would require accurate "ranged cold" attacks.
Patch/Close a door being broken down to hold your defensive fortification longer.
I'm sure dozens of other things could be figured out as well, but this shows it has general uses that are fairly utilitarian. Even if these have overlap with other elements, that just helps to how that they each have viability.
Interestingly, because of the ability to add weight to things, you could make someone else's shield, weapon, etc extremely heavy by adding lots of ice to it. Because of that you could easily disarm opponents or reduce their ability to defend themselves.
Similar to the above, freezing their feet in place is pretty alright (assuming they're not so strong to lift their feet anyway.
Freezing their helmet closed and fusing it to their armor could quite possibly suffocate them.