Using the word 'kind' for a category

I found on englishplus (accessed on 23 Dec 2012) that the phrase ‘kind of animal‘ refers to a category of animal.

If you are using an expression like kind of, sort of, type of, or variety of, then you are putting the person or thing you describe into a category. Therefore, such expressions should not apply to one specific individual of the type.

Incorrect: He is the kind of leader we need.
(You are referring to a specific person in the subject and to a category in the predicate.)

Correct: He is like the kind of leader we need.
(You are referring to a category, and he is an example of someone in the category.)

The confusion I have is that I feel now it is inappropriate to say “Spot is a kind of dog” (where Spot is a proper noun and the name of a certain dog). Instead, one should say “Spot belongs to a kind of dog” or “Spot is of a kind of dog”. The idea that “Spot is a kind of dog” is incorrect follows from “He is like the kind of leader we need” being said correct by the above reference.

Then, I referred to the OALD. Its first sense for the noun form is: “a group of people or things that are the same in some way…”. To me it seems to validate the explanation at englishplus.com. But then it gives the following examples, among others:

1. The school is the first of its kind …
2. They sell all kinds of things.

(1) seems to match with the theory that kind is a group.
(2) seems to suggest that kind is not a category but perhaps a common noun – since it is ‘sell’able.

In the kind of animal phrase, I assume that of animal is a prepositional phrase acting as a qualifier of the noun kind used to select the kind required. Therefore to say that I am referring to a category of animals, I use the phrase kind of animal and then to show that T is an animal of this category, I say: “T is an animal of a {kind of animal}” (ie, T is an animal of a kind of animal). Now, I have never come across such a sentence in my life. Please help me understand the source of the confusion and the correct usage of the word kind, whether as an idiom or by proper grammar.