Any infinite cardinal that is also an ordinal is an infinite well-ordered cardinal. Any cardinal of the form aleph a where a is an ordinal. Every infinite well-ordered cardinal is an infinite cardinal. In ZF it is possible to have an infinite cardinal which is not a well-ordered cardinal. It is the cardinal successor of aleph 0 .
Every successor cardinal is a well-ordered cardinal.
Every well-ordered cardinal that is not a successor cardinal is a limit cardinal. Any cardinal of the form aleph a , where a is a limit ordinal, is a limit cardinal. The inconsistency is about whether to consider 0 and aleph 0 limit cardinals. The reason that I prefer the definition I give is that it makes the partitions of cardinals much nicer, since we do not need to consider 0 and aleph 0 special cases, i.e., well-ordered cardinals which are neither successors nor limits. Also, with the definition I use aleph 0 is considered a regular limit cardinal and a regular strong limit cardinal.