Are there 3 definitions of rectangular numbers?

Question

I'm a little bit confused about the definition on rectangular numbers, which is also giving me doubts about its function (or idea).

1. So there's the definition of that all rectangular can be found with this formula: $$ f(n) = (n-1) \cdot (n+4) $$ We get: 0, 6, 14, 24, 36, 50.

2. Others define it as the double of triangular numbers: $$ f(n) = n \cdot (n+1) $$ Then we get: 0, 2, 6, 12, 20

3. There are also numbers that are neither primes nor squares, like: 8, 10, 16, 18, 22. These numbers can be graphically displayed as rectangles of dots. Or do we call these composite numbers?

So I'm a little bit confused here. I'm also very interested in what their uses are. Does anyone have an answer?