The designator child represents the oldest child (note) of the note currently in focus, thus the first as listed in Outline view. Mention in various references of 'oldest' deriving from creation/modification date are erroneous. 'Oldest' equates to the note whose $OutlineOrder attribute value is 1 greater than the current note's OutlineOrder value.
The logical opposite of this (first) child designator is lastChild; there is no firstChild designator.
Designators don't allow offset references as with attributes. Thus $Name("Some note") works but child("Some note") doesn't. If an offset designator is needed use find() instead.