Terminology is tricky. To make it worse, I'm not much of a baker, less of a pastry maker, and certainly no pastry chef -- whether in the past, now or ever.
Icing can be done with many things, including glaze. Icing is what you do and what you use to cover a baked product with a sweet substance. "Icing" can be, but is not always synonymous with "glaze." As I understand the term, icing also includes frosting.
In my earlier post, when I termed glaze as "thin," I meant: As a finished coating on the final product, a glaze will end up only a fraction of an inch thick. As a texture, once cooled, glazes are usually termed "stiff" or even "brittle." You may be conflating "stiff" with "thick," which is legitimate. I was not, which may have been confusing. To make it even more confusing, syrup and sugar glazes are often applied hot when they're thinnest. On the other hand, egg washes which are also glazes, are thinnest when they're cool and before baking.
Cake icings are broken into several categories. "Frostings" are the fluffiest. Glazes lie the flattest. The "thinnest" glazes I know of are egg washes and syrups. Fondants are not glazes, they are very thin frostings. Glazes are (almost always) applied as liquid. Cakes with holes in the middle, such as bundts, are a PITA to frost. Perhaps that's why glazes predominate as icings.
My head hurts,