My guess is a system such as :
$a_{11}x_1+a_{12}x_2+...+a_{1n}x_n=b_1$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ a_{22}x_2+...+a_{2n}x_n=b_2$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a_{mn}x_n=b_{m}$
i.e. a system $Ax=b$ where $A$ is upper triangular. Would we call the system triangular even if $A$ is lower triangular? I think we should.
Yes it is an (upper) triangular system which can be obtained for any linear system by Gauss elimination algorithm.