zoology - Species with reproduction barriers that can both reproduce with a third species

To start with, I do not have a sound knowledge in biology or any formal education in the area.

I was told that one of the definition of a species is a reproductive barrier, which means that if two animals can't reproduce, they are of different species. The barrier can be either the inability of the sperm to fertilize an egg, or a physical trait that inhibits reproduction, e.g. a cricket species that has a different mating song than another species or two species of flies, one that mates on yellow flowers, and the other on red flowers.

But what happens when, while two species can't reproduce, but there is a "chain" of "intermediate" sexual partners that can produce reproductive connection step by step. Like 6 degrees to Kavin Bacon, but with animal sex.

I'll try to explain with an example:

A Great Dane and a Miniature Pinscher dogs can't mate due to obvious size differences. But The Pincher can mate with a German Pincher (a slightly bigger breed of Pincher), which can mate with a Doberman Pincher. And the Doberman can mate with a German Shepard which can mate with a Great Dane.

I've also heard that such things happen with birds and crickets, where there is an original species, from which evolved several other, and while the original species (which still exists) can mate with all the new species. Some new species can't mate with some, or all of the other new species.

How are such species are defined, and at what point dogs stop being dogs anymore?