Just like the meaning of the word Emergency, only in necessary impromptu situations can the money be touched. A different reason why most people make mistakes is that they don't really know what emergency is all about. Aside having emergency funds kept, there should be extra side of discretionary income kept to foot minor bills that may not have come during calculating expenses.
Emergency funds can be consolidated, the heavier the funds the better effective against emergency that should warrant affecting your investment.
There's a vital point here just as you have said many find it's hard to know what emergency fund is, let me use the word unforseen circumstances, thing that one can't think off or expected after budget, as the emergency or unforseen circumstances don't happen at every give time and may happen when one don't expect it , it's very hard to be capture or quantify the certain amount for such unforseen to keep fund outside make it safe for relief if it happens. A good planner don't see any Mino issue to be unforseen where emergency fund could be spend because those Mino could even be cover by its budget, what many called emergency today is as a result of poor planning and inadequate budgeting .