If I recall correctly, the Rawlsian problem is not maximizing the welfare of the worst off.  Rather,in its most strict form, it is maximizing the sum of everyone's welfare, such that (1) no one is made worse off and (2) the least well-off see some increase in welfare.  A less restrictive Rawlsian specification relaxes the first condition, but only when there are no possibilities for improving the welfare of the least without reducing the welfare of others.  That is, it's not fair to take from the rich to give to the poor unless it's the only possible way to avoid reducing the well-being of the poor.

This is a bit different from simply maximizing the welfare of the least well off and makes pinning down what we mean by the "quasi" part of your definition a bit more important.