Here are some possible definitions you might consider using.

Class: A concentration of unusually high probability density in Thingspace.

Type: A subclass. An even denser area of thingspace or conceptspace within a cluster of things.

Metric: a scale you use to measure a single trait of something. In humans, that could be height, weight, hair color, etc. In order to be useful, a metric must give you further information about that thing as opposed to other things in its class/type (there must be significantly more variability along that dimension than others, in terms of thingspace).

In regards to the article itself, it highlights the difficulty of projecting a multidimensional space (with the number of dimensions equal to the number of metrics you're using) and a complex distribution of "goodness" within that space to a single dimension of goodness with minimal complexity and minimal loss of information.