The original post may have some ambiguity in the term "minimum" but I read it as the minimum of the current contents of the stack, based on where the link the OP provided went to. When I said "minimum to date", I meant relative to the current contents of the stack in order. That is, if you push a value larger than the minimum on the top of the stack, you copy the minimum, if it's smaller, the pushed entry on the stack will be the pushed value, then the pushed value again as the minimum. This always keeps the minimum of the stack on the top of the stack.

"All time minimum" would just be one place to store the minimum, since popping would do nothing to it. It is equivalent to the simplest algorithm to find the min of an unsort array/set/whatever.