What Is Entropy?

My understanding is that entropy is a way of quantifying how many different ways a thing could 'actually be' and yet still 'appear to be' how it is. So it is largely a result of an observer's limited ability to perceive / interrogate the 'true' nature of the system in question.

So for example you could observe that a single coin flip is heads, and entropy will help you quantify how many different ways that could have come to pass. e.g. is it a fair coin, a weighted coin, a coin with two head faces, etc. All these possibilities increase the entropy of the system. An arrangement _not_ counted towards the system's entropy is the arrangement where the coin has no heads face, only ever comes up tails, etc.

Related, my intuition about the observation that entropy tends to increase is that it's purely a result of more likely things happening more often on average.

Would be delighted if anyone wanted to correct either of these intuitions.

When ice cubes in a glass of water slowly melt, and the temperature of the liquid water decreases, where does the limited ability of an observer come into play?

It seems to me that two things in this scenario are true:

1) The fundamental physical interactions (i.e. particle collisions) are all time-reversible, and no observer of any one such interaction would be able to tell which directly time is flowing.

2) The states of the overall system are not time-reversible.

For example: Say I'm at some distance from you, between 0 and 1 km (all equiprobable). Now I switch to being 10x as far away. This is time-reversible, but because the volume of the set of states changed, the differential entropy changes. This is the kind of thing that happens in time-reversible continuous systems that can't happen in time-reversible discrete systems.