Does CSL contain an empty string or not? Is empty string accepted by LBA or not?
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I only know of the latter definition, i.e. the one allowing $S \to \varepsilon$ but disallowing $S$ in the right sides of productions.
Ultimately this is done to have a nice definition in the sense that the class of context-sensitive languages forms a proper superset of the class of context-free languages.
Definitions excluding the empty string to be in context-sensitive languages are a little bit less nice, in that sense.
As you found out, definitions may vary between sources and are often chosen in order to state some property or result in a particularly nice manner.
This is common in mathematics and another example would be the question of including 0 in the natural numbers $\mathbb N$ or not.
The thing is that usually these little changes do not really matter in the end; they do not fundamentally change the mathematical nature of the thing defined, and this is also the case with the small differences between the definitions for context-sensitive languages you encountered.