I am working from Munkres' Analysis, and I've converted his definitions slightly to make them easier to compare. In the table below, you can fill in the blanks in the top row with words from the lower rows to form either definition:
It is not clear to me what motivates some of the choices when it comes to 'filling in the blanks'. My biggest concern is the last two blanks. The 2nd blank is essentially discussed in my old question here:
And while I roughly understand the response there (letting in isolated points means that functions can approach infinitely many limits at isolated points), when I consider changes to the last two columns, I find myself also considering changes to the 2nd.
My hope is that someone can construct simple examples for each column (in as few dimensions as possible!) which motivate the choice, while somehow dealing with the interconnection problem wherein choice in one column affects choice in another...