Compared to a regular upper triangular matrix, the upper triangular matrix $R$ obtained from the regular matrix decomposition methods has the following additional property:
$R$ has a shape so that every row the first non-zero element appears at least one column later than in the preceding row.
I.e. the regular decompositions would never return an $R$ like this: $$ \pmatrix{0&1\\0&1} $$
And instead return $$ \pmatrix{0&1\\0&0} $$
Of cause, I can't pledge that all implementations of all matrix decompositions actually obey this rule. But so far, all I've checked did.
What is the name for an upper triangular matrix with the special property described above?
It's an upper triangle matrix in row echelon form.