Is Semantic Preservation Soundness (or Correctness) or Completeness
This is really a question about the notion of operational correspondence. 

The paper Towards a Unified Approach to Encodability and
Separation Results for Process Calculi by Daniele Gorla (http://wwwusers.di.uniroma1.it/~gorla/papers/G-CONCUR08.pdf) is concerned with correctness criteria for translations between process calculi. 

In his work, Gorla introduces a notion of behavioural equivalence; I will leave it out here in order not to complicate the explanation overly.

Let $[ \! [ \; ] \! ]$ be an encoding, let $\rightarrow_1$ be the transition relation defined for the source language and let $\rightarrow_2$ be the transition relation defined for the target language. If we have that


If $S \rightarrow_1 S'$ then $[ \! [ S ] \! ] \rightarrow^*_2 [ \! [ S' ] \! ]$, we say that the encoding is complete
If $[ \! [ S ] \! ] \rightarrow^*_2 T$, then there exists a $S'$ such that $S \rightarrow^*_1 S'$ and $T \rightarrow^*_2 [ \! [ S' ] \! ]$, we say that the encoding is sound.


So yes, the notion mentioned in the questions is indeed that of completeness. The notion of soundness is shaped by the fact that the target language will often be richer than the target language or contain syntactic constructs whose behaviour does not correspond to anything in the source language. This explains why we speak of the intermediate configuration $T$ in the definition instead of requiring injectivity.