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The International Vocabulary of Metrology defines a kind of quantity as the following:
However, to me NOTE 1 EXAMPLE 1 and NOTE 1 EXAMPLE 2 are contradictory with NOTE 2 EXAMPLE.
NOTE 2 EXAMPLE gives me the idea of kind of quantities being "flavors of dimensions", but the examples of NOTE 1 are then saying the exact contrary, with diameter, circumferences and wavelength being of the same kind.
Could someone clarify what the relationships between a quantity, a quantity kind, and a dimension?
They aren't "kind of quantities" they are "kinds of quantities". One kind of quantity is lengths. Another is energies.
You could substitute "type" for "kind" and say that height and width are two quantities of the same type, instead of quantities of the same kind. The only reason not to is some standards body has chosen to arbitrarily standardize on using the word "kind" instead of "type" for this usage.
Height and wavelength are quantities of the same kind because we can compare them to each other: the wavelength of this wave is greater or less than the height of some object.
The thermal energy in this ball is a quantity of the same kind as the kinetic energy of the ball, because we can compare them and decide if one is more or less than the other.
Could someone clarify what the relationships between a quantity, a quantity kind, and a dimension?
Quantity is defined in your linked document as "property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference"
To put it more simply it's something you can measure (at least in principle). The height of a building. The kinetic energy of a ball. The temperature of a water bath.
A Kind of Quantity is a group of quantities that can be compared with each other.
We can say that the wavelength of red light is less than the height of some building. So these two quantities are quantities of the same kind.
We can't say that the kinetic energy of the ball is more or less than the weight of a particular blue whale. So these are not quantities of the same kind.
Dimensions are trickier to define. Your document defines dimension as "expression of the dependence of a quantity on the base quantities of a system of quantities as a product of powers of factors corresponding to the base quantities, omitting any numerical factor"
This basically means if we measure two things with the same units, they have the same dimensions. Using their example, the energy unit joule is formally newtons time meters. Similarly, a moment of force is measured in newton-meters. So an energy quantity and a moment of force quantity have the same dimensions. But because we can't directly compare energies with moments of force, we wouldn't say these two quantities are the same kind.
