Is this a contradiction in Wikipedia's definition of temperate zone?
Note that they say:


  the temperate or tepid climates of Earth occur in the middle latitudes


The word in can be used to convey that the first thing is only a part of the larger second thing... much as, for example, Texas is found IN the United States, or the preteen years are found in childhood. 

But by including the additional further explanation of what the middle latitudes are directly after that, I agree it leans a reader towards your interpretation, and is ambiguous and likely confusing to others.

I'll try to make some minor adjustments to the article to hopefully alleviate any confusion.

But in all honesty, it's probably better to not get overly caught up in such excessively precise categorizing.  I tend to believe that what one person, even one climatologist, might call temperate climates may well vary from what another one would classify them.
For example, they can be based on temperature ranges, on prevailing wind patterns, on breaking into equally-sized regions (like Fred commented), etc.

Really, I'd think of the subtropics as sort of a blurry boundary between the tropics and the temperate zone... where the closer you get towards the Equator, the less seasonal variability and frequency of cold air intrusions.

Indeed, even typical places well across into the "official Tropics" can see occasional years with some notable cold fronts, as happened in Cuba and even Costa Rica in the winter of 2010.  So it's not like 23.5 is some magic physical boundary just because that's the northern extent of the sun's latitudinal cycle. 

 

So the Wiki article as you read it appears to be saying the temperate climates are only 35° and 66.5°, only a subsection of the middle latitudes.  But honestly, exactly what you classify the boundaries as is sort of like arguing over which states are in the Midwest or Southern Southern US -- very unnecessary outside of very specific circumstances, and prone to at least occasional dispute probably even among experts.  Yes, a teacher or test may ask you to know their specific definition (whether of temperate boundaries or of which states are in the Midwest)... that's just part of life sometimes.

But the real take-home idea towards understanding how the Earth and specifically climate/weather work is just that there's a Polar region which has very cold polar air basically year-round, a Tropical region which has very warm Tropical air year-round (outside of colder mountains!), and then a "battle ground" that shifts seasonally and even daily in between, altering the weather significantly in those locations.  And local factors like bodies of water and wind regimes can significantly shift regions into a large array of different expected weather