From what I can tell from the definitions of a lower-triangular, upper-triangular, and diagonal matrices, I've come to the conclusion that the zero matrix is in the set of all of each type of matrix. Is this correct?
A zero square matrix is lower triangular, upper triangular, and also diagonal.
If an object meets the definition of three things then it is the three things. What are you confused about?
Provided it is a square matrix.
An upper triangular matrix is one in which all entries below main diagonal are zero. Clearly this is satisfied.
An lower triangular matrix is one in which all entries above main diagonal are zero. Clearly this is satisfied.
An diagonal matrix is one in which all non-diagonal entries are zero. Clearly this is also satisfied.
Hence, a zero square matrix is upper and lower triangular as well as diagonal matrix.