Need help in understanding the editorial for MINIMISING COINS (CSES)

General Questions
#1

I was solving minimising coins problem on CSES. I was successfully able to solve.
My solution has two for loops.
The outer for loop runs over the sums from 1 to x
The inner loop iterates over coins

	vector<int> dp(x + 1);
	dp[0] = 0;
	for (int i = 1; i <= x; i++)
	{
		int ans = INT_MAX;
		for (int coin : coins)
		{
			if (i - coin >= 0 and dp[i - coin] >= 0)
			{
				ans = min(ans, 1 + dp[i - coin]);
			}
		}
		dp[i] = (ans == INT_MAX ? -1 : ans);
	}
 
	cout << dp[x] << endl;

But in the editorial on USACO guide

The outer for loop runs over the coins
The inner loop iterates over target sum from 0 to x

int main() {
	int n, x;
	cin >> n >> x;
	vi coins(n);
	for (int i = 0; i < n; i++) { cin >> coins[i]; }
	for (int i = 0; i <= x; i++) { dp[i] = INT_MAX; }
	dp[0] = 0;
	for (int i = 1; i <= n; i++) {
		for (int weight = 0; weight <= x; weight++) {
			if (weight - coins[i - 1] >= 0) {
				dp[weight] = min(dp[weight], dp[weight - coins[i - 1]] + 1);
			}
		}
	}
	cout << (dp[x] == INT_MAX ? -1 : dp[x]) << '\n';
}

Can someone help me in wrapping my head around the second approach. I am unable to figure how can the second approach also work correctly.