It is given here in Theorem 3.3, that if $a \equiv b \pmod n$; then $b = a +nq, \exists q \in \mathbb{Z}$.
I feel that is more logical to state otherwise (i.e., $a = b + nq$), as is more close to euclidean division algorithm.
The article chooses the opposite notation for ease in theorem proving, but seems weird to me, although the author justifies this by symmetry property of modulus arithmetic earlier. I feel the author's syntax is not easy to use without justification (i.e., of symmetry property of congruence arithmetic) before.
However, later the author in Theorem 3.4 takes a general view of $a,b$ as congruent quantities w.r.t. the modulus.
It leads to transformation between congruence relation and equality : $a \equiv b \pmod m \iff a \pmod m = b \pmod m$.
Logically the two statements are equivalent due to the symmetry of congruence mod $(n)$.
Therefore, we can not make a logical decision on which one is a better choice. The author has made a decision based on the flow of his proof, so we may accept his/her choice.