Difference between change (n) and coins (n)

Solution 1:

The definition of change that was provided is incomplete:

[Merriam-Webster]
2 a : money in small denominations received in exchange for an equivalent sum in larger denominations
2 b : money returned when a payment exceeds the amount due
// a cashier quick at making change
2 c : coins especially of low denominations
// a pocketful of change

Only sense 2 c was included as the definition in the question.

Coins represent money, and they can always be considered change in the one sense of their definition (2 c above), but in the other two senses of the word, coins are not always what's returned.

In other words, change (in the other senses of the word) can be either bills or coins:

She was given small bills and coins in change.

So, in some cases, the words can be used interchangeably—but not in all cases. It depends on the context.