One of the ways the Most Noble and Ancient house of Black keeps their royal power above the Ministry of Magic, is that royal children don’t have the trace. This keeps all their magic in general hidden from the Ministry as the trace can detect any magic cast near kids who have it.

So basically in canon pure-bloods are defined as are wizards whose parents are either pure-blood or half-blood and whose grandparents are either pure-blood or half-blood. So I surmised that muggle-borns must be wizards whose parents are muggles and whose grandparents are either muggles or squibs.

Family names are passed through the male line, so a muggle-born who had a squib ancestor could be born into a pure-blood family and have a pure-blood family name. So in other words, a pure-blood family could produce a muggle-born, known as a bluemud.

In the case of Pegasus and Hugo, Pegasus is a squib born into a pure-blood family, the Black family. Hugo legally took Pegasus’s last name, Black, so her family status passed to their sons who were muggles.

This is how Hermione who had muggle parents but a squib for a grandparent can be both a muggle-born and a member of the House of Black.