Definition of minimal phase
these definitions do not make it clear if a system with one pole or one zero which has a real part of zero is still called minimal phase or not.


First, from the Wikipedia article Minimum Phase:


  
    a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
  


And from the article BIBO Stability:


  
    For a rational and continuous-time system ... all poles of the system must be in the strict left half of the s-plane for BIBO stability.
  


It makes sense that a system with a pole on the imaginary axis is not considered stable, since that pole would mean that a bounded input (a sinusoid) would produce an unbounded output.