Thomas (or Alan?) de Mumby, of Mumby - Willoughby/Mumby conflation?

If you print his decendency chart, you see Thomas/Alan de Mumby as father of William married to Alice de Beke AND as father of Agnes married to William de Holtoft.
Agnes and William de Holtoft are also are given as the parents of Agnes Willoughby and William de Holtoft
Agnes is also given as the child of Thomas de Mumby/Willoughby
The print out would look like this (if it stays aligned)

Thomas (or Alan?) de Mumby
___________________________|___________________________________________
| | |
William de Mumby,---Alice de Becke Agnes de Mumby----William de Holtoft Sarah de Mumby
| |
Thomas de Mumby---Margaret de Mumby Agnes Willoughby----William de Holtoft
AKA Willoughby |
|
Agnes Willoughby----William de Holtoft

It seems to me that the Agnes's and William's have become confused.

Consider the following
https://www.google.com/books/edition/A_Genealogical_and_Heraldic_Hi...
A Genealogical and Heraldic History of the Extinct and Dormant Baronetcies ...
By John Burke
pg 497 footnote
By ALICE his wife sister and sole heir of Ralph de Muer Lord of Calthorp and Covenham in the county of Lincoln SIR WILLIAM DE HILTOFT was son of another Sir William by Agnes daughter of Thomas and one of the sisters and co heirs of William de Mumby a family descended from the house of Willoughby. SIR WILLIAM DE WILLOUGH BY having two sons by his wife ALICE BEKE heir of the feudal lords of Eresby. The elder, Robert, inherited the barony on the decease of his uncle Anthony Beke, bishop of Durham, and was summoned to parliament as Lord Willoughby de Eresby. The younger Thomas DE WILLOUGHBY m Margaret, sister and co heir of Alun de Munby, and took the name; his son William dying without issue, the estates passed to Sir William Hiltoft, the husband of his daughter Agnes.

I think i can show this best starting with the youngest. If it doesn't' stay aligned, you should be able to figure it out from the above footnote.

William Willoughby
|
Thomas Willoughby---
William Hiltoft | who took his wife's | Alice de Beke
William Hiltoft --| | surname Thomas de Mumby
| | |
| Agnes de Mumby---- William de Mumby--
| |
| Margaret de Mumby ---

Have I interpreted Burke correctly?
Does the second pedigree seem more likely?