Reduced row echelon form

A matrix A is said to be in reduced row echelon form iff:
a) every leading non-zero entry is to the right of the leading non-zero entry in the row above.
b) every leading entry in each non-zero row is equal to one.
c) in every column with a pivot, the pivot is the only non-zero entry in that column.
d) all zero rows are grouped at the bottom of the matrix.

An example of a matrix in reduced row echelon form(RREF):
|1 0 0 0|
|0 1 0 0|
|0 0 0 0|
|0 0 0 0|

The rank of a matrix is the number of rows in the RREF of the matrix. In the example the rank of the matrix is 2.

There appears to be some conflict in terms of what the RREF(Row Reduced Echelon Form) of a matrix actually is.

Following are two different forms, both of which are commonly referred to as RREF:

Form 1

|* * * *|
|0 * * *|
|0 0 * *|

Where * is a number.
This is what I was taught to refer to as RREF, however someone recently informed me that this form is actually called Row Reduced Form (thanks 10998521).

It is simple to stop here, as simple division will give you the result for the bottom row, the inserting that result into the row above and solving will give you the solution for the second row, and so on (this is known as back substitution) (essentially, it assumes that it is easier to do simple algebra than to work with elementary row operations).

Form 2

|1 0 0 *|
|0 1 0 *|
|0 0 1 *|

Where * is a number. This is what I was taught to refer to as Gauss-Jordan form, but it is also commonly referred to as RREF. It may not always be possible to achieve this form, but it is easier to read, as the solutions for each row are directly written in it.

On a side note, the echelon in RREF refers to the triangle shape that can be drawn around the zeroes in the corners.