What defines one "season" of anime? From what I've been able to follow, the seasons of anime shows have been tied to thirteen week blocks, directly correlating with the actual season in Japan (for instance, this is the Winter 2013 season).

However, there are instances in which shows with more than 13 episodes are considered to have only run for one season.

Which is the correct definition - are seasons 23-26 episodes, or are they 11-13 episodes?
The 13 week "season" that refers to a time of year or when a show starts airing is sometimes referred to as an "airing" season or a "television" season. This is like "Winter 2013 season". But since shows sometime span different television seasons, even if it's 13 episodes long, this run of a series can also be referred to as a "season". 

Sometimes shows get cut off, go on hiatus, or for whatever reasons (e.g. financial) have a significant break between certain episodes such that the TV series runs through multiple "television seasons" (e.g. Girls Und Panzer) yet the show itself is still referred to as the "first season", and this is sometimes referred to as a "cour" or "series". The usage of "season" comes down to semantics. Multiple "seasons" of a TV series can also be called "series", like Galaxy Angel and it's multiple (different lengths) "seasons" are sometimes called "series 1", "series 2", etc. "Cours" however, usually refer to a 12 or 13 stretch of episodes. So a 26 episode series can be referred to as "2 cours".

But as far as the length of an anime "season", a 12 episode series run can be referred to as a "season", yet sometimes a 26, or 36, or 201 (Gintama) episode run is called a "season". It's technically not tied to a specific length, which is why some people prefer to use "cours".