What Are the Ideas Behind Variations of the Coin Change Problem?
OP here, answering my own question for the greater good of mankind.

Let dp[i][j] denote the solution to the [i,j]-th subproblem; that is, dp[i][j] is the number of ways to make change for the amount i, using coins j through n.

Coin Change Problem with Repetition

dp[i - xs[j - 1]][j] + // 1
    dp[i][j - 1]       // 2



Use coin j, and since there is no constraint on the denominations, solve the smaller subproblem using the same denomination.
Don't use coin j.


Coin Change Problem without Repetition

dp[i - xs[j - 1]][j - 1] + // 1
    dp[i][j - 1]           // 2



Use coin j, and since we used coin j for this subproblem, we can't use it for any other.
Don't use coin j.