The dual certification question is quite different from the variable penetration problem.
The straight and L grade compositions overlap, and it is possible to meet the higher strength requirements of the straight grade with carbon content at the upper end of the L grade range. Dual certification of these grades is mainly driven by commercial imperative; i.e., reducing stock for jobbers. Usually it is not a problem; with normal welding practices, risk of sensitization with the dual certified grades is minimal. However for high temperature applications you normally would want to specify 304H.
As far as weldability is concerned, you can mix and match 304/304L, 316/316L and their respective matching filler metals without consequence (although I don't recommended doing that). Your welders won't notice a difference.
Corrosion-wise, 316/316L offers improved chloride pitting resistance over 304/304L, and is typically the first step up the alloy ladder when 304/304L is inadequate. However 316/316L offers no improvement in chloride SCC resistance.
"Everyone is entitled to their own opinions, but they are not entitled to their own facts."