Alice de Lacy (de Madeville/de Vere) - Alice de Lacy's SURNAME

i am very confused as to why Alice De Lacy, born of a father whose last name is MANDEVILLE,
isn't named Alice Mandeville.

She in turn marries John FitzRichard de Lacy, who himself isn't a Lacy. He is John fitz ("son of") Richard,
who in turn is Richard fitz Eustace, who in turn is Eustace fitz John de Burgh.

Two things confuse me utterly:

1. All of the bloodline following the marriage of John FitzRichard de Lacy and Alice would not
actually be Lacys at all -- or so it seems to me. They would actually be "de Burghs".

2. When and where does Lacy come into the picture? It happens somewhere between
John de Burgh (1060-1138) and and Alice Mandeville (1150-1248) but how and why?

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Being name Lacy, myself, I am not interested in know any adoptive line in my past. I'm wanting
to know my BLOOD line. If I am actually a "de Burgh," fine, but I want to know that.