Does normalized Latin rectangle have to have its first column in a 1, 2, 3, ..., n or can it be for example 1, 3, 4, ... n aka in just ascending order.
By example, is $\begin{pmatrix} 1&2&3&4 \\ 3&4&1&2 \end{pmatrix} $ a normalised Latin rectangle or it has to be $\begin{pmatrix} 1&2&3&4 \\ 2&... \end{pmatrix} $ ?
You can encounter "normalized" as meaning either of these things, neither of which are standard. Which definition is being used needs to be indicated somehow in the source material.
In my papers, "normalized" refers only to the first row in order, while "reduced" refers to having the first row and first column in order. It'll be explicitly defined somewhere in each paper. However, I've seen "normalized" and "regular(ized)" and "standard(ized)" used in various ways in various papers.
If it's not defined, you might need to work with both definitions until you find an inconsistency indicating which definition is being used.