Why can't we store light in the form of light?

For the photons that make up light to exist they have to be travelling at the speed of light. This means that to store them you have to put them in a container where they can move around at the speed of light until you want to let them out.

You could build the container out of mirrors, but no mirror we can currently build is 100% reflective, or indeed can be ever 100% reflective. Usually when a photon "hits" the mirror it is absorbed by one of the atoms in the mirror and then re-emitted back out into the container. However, occasionally the photon either won't get re-emitted (leaving the atom in an excited state) or it doesn't hit one of the atoms and makes it way through the mirror and out of the container.

While the chances of this happening for an individual photon are low, there are lots of photons travelling very fast so it happens many times thus causing the light to "leak" or decay.

Building a near perfect mirror is hard, so it's easier to convert the light into something that can be stored and then convert that back into light when you need it.

Your examples are a bit misleading. For example you say:

We can store cold (ice),heat (i.e. hot water bag)

But we can only store heat temporarily, just as we can only store light temporarily. Your ice pack will eventually heat up and your hot water bottle will eventually cool down, just as light stored between two mirrors will eventually escape.

and electrical charge (batteries)

Charge isn't stored as charge in a battery. A chemical reaction generates the charge. This would be the same as converting the light to something else, storing that something else then regenerating the light when needed.

We can even "store" a magnetic field in a magnet.

Not the same thing, as we are not storing magnetic charge in a magnet.

A method of storage that might just fit your criteria is storing light in a Bose Einstein condensate. Light pulses can be brought to a halt in a BEC, and in principle stored indefinitely.

I'll try to answer your question in the spirit of how you asked it. Basically you can't really store anything you've mentioned. The ice will eventually heat up, the heat will cool down and the battery will lose charge. A box of mirrors with light shined in it will "store" light like your other examples but it will lose the energy much faster then any of them. Imagine throwing a superball into a box and closing the lid, does it bounce forever and "store" the energy?