Your definition of the Heaviside step function has H(0) = 1. Your definition of L has L(0) = 1⁄2, so you’re not really taking the derivative of the same function.
I don’t really believe nonstandard analysis helps us differentiate the Heaviside step function. You have found a function that is quite a lot like the step function and shown that it has a derivative (maybe), but I would need to be convinced that all functions have the same derivative to be convinced that something meaningful is going on. (And since all your derivatives have different values, this seems like a not useful definition of a derivative)
Your definition of the Heaviside step function has H(0) = 1.
Your definition of L has L(0) = 1⁄2, so you’re not really taking the derivative of the same function.
I don’t really believe nonstandard analysis helps us differentiate the Heaviside step function. You have found a function that is quite a lot like the step function and shown that it has a derivative (maybe), but I would need to be convinced that all functions have the same derivative to be convinced that something meaningful is going on. (And since all your derivatives have different values, this seems like a not useful definition of a derivative)