Why are $L$-reductions defined the way they are?
L-reductions were defined by Papadimitriou and Yannakakis in their paper Optimization, approximation, and complexity classes. From the abstract:

Furthermore, we show that a number of common optimization problems are complete for MAXSNP under a kind of careful transformation (called L-reduction) that preserves
approximability.

A survey of various related notions of reducibility is Ausiello and Paschos, Approximability preserving reduction. Each of the notions of reductions appearing there was invented to serve some purpose. Definitions are typically not arbitrary. They are supposed to model some intuitive notion. Some definitions are more successful than others. When unsure about a definition, it's always a good idea to check out the original paper(s) introducing the definition.