Different definitions of unit step signal
The Wikipedia article on Heaviside has a section that discusses the 3 different conventions on $H(0)$:

https://en.wikipedia.org/wiki/Heaviside_step_function

Essentially, the step function is used in probability theory as well but is called something different.

I believe that the "symmetric" $H(0)=\frac{1}{2}$ version is probably most suitable in Linear Systems Theory. In discussions on the Gibbs phenomena, the truncated Fourier series interpolates discontinuities at the mid-value which in this case is $\frac{1}{2}$.  

Also, if you look at the article, there are a number of limits of continuous functions like Erf, that approach the step as some parameter approaches some value. If you were to set up a physical experiment, you would never be able to realize an ideal step function, you would need infinite bandwidth, so in reality, the best you could achieve would be something like a one of those continuous functions.  $H(0)=1/2$ is a vestige of physical realizability.