A deeper, physics/chemistry approach to the question:
A battery (whatever chemistry it uses) invariably contains solid electrodes and liquid electrolyte (there are batteries that have it the other way round, but the problems are pretty much the same).
The dust-sized particles that the electrode is made of, contain some substance in their entire volume, but the chemical processes of charging and discharging happen only at the interface between the solid particle and the liquid electrolyte.
No matter if you charge or discharge the battery, the surface of the particle gets enriched in the reaction product and depleted of the substance yet to be processed. The process that transports the product to the inside of the particle and the unprocessed substance to the surface is called diffusion.
The diffusion is profoundly temperature-dependent. Every few degrees C double its speed.
It is also concentration-dependent - if there is more unprocessed mass inside the particle, the diffusion will go faster.
If you force the battery to do its job faster (use higher charge or discharge current) the diffusion may fail to keep up. This manifests as a non-linear electrical resistance of the whole cell, limiting the usable power.
If we combine this with the above inherent diffusion properties, we see that the available power depends on both the temperature and the state of charge of the battery. If the temperature is low, the available power capability of the battery falls sooner below whatever usable power one needs.
- at 30C the battery may be happy at its design capacity.
- at 15C the battery may be usable between 5% and 95% state of charge, giving us ~90% of usable capacity
- at 0C the battery may be usable between 20% and 80% so we have ~60% in between.
... and so on.
When charging the battery in a cold weather, we have the luxury either use some extra energy to heat the battery up so it could charge at a high rate all the way to 100% - or - use lower charge rate at the end of the process, reaching 100% e.g. overnight.
On the other hand, when discharging the battery, e.g. driving an EV car or talking on the phone, the power demand is generally dictated by the load and if the battery cannot meet it, it is considered empty even if it could give off e.g. 30% more energy at lower rate.
Two more interesting battery properties, also related to diffusion:
The electrolyte generally has the same diffusion-related behavior, but it happens at different timescale (e.g. seconds). This is why a starter battery (found in cars) can give off 1-3 kW for few seconds (needed to start the engine) but needs some "rest" if the engine fails to start and one needs a second attempt. The needed "rest" also depends on the temperature and in sub-freezing temperatures a minute or two is advisable.
The problem with the solid-state diffusion (and the temperature dependence of the usable capacity) gets worse as the battery ages. This is because the particles in the electrodes tend to get bigger with time and the diffusion has to carry the substances over greater distances.