Diego Lainez - Medlands

Erica Howton maybe this will become clearer to me tomorrow after I've rested, but where I'm still stumped is that this is the only mention I found for Diego Lainez on the Medlands link shared:

https://fmg.ac/Projects/MedLands/SPANISH%20NOBILITY%20EARLY%20MEDIE...

JIMENA Díaz (-1106). The "Corónicas" Navarras record that "este meo Çid" married "dona Xemena, nieta del rey don Alfonsso, filla del conte don Diago de Asturias"[510]. "Ruderigo Didaz" accepted as his wife "Scemena filia Didago ducis de terra Asturiense" and granted property "in territorio Kastelle…in Cavia" to her, including "villas que michi saccarunt Albaro Faniz et Albaro Albariz sobrinis meis", with "comes Petro Assuriz et comes Garsea Ordonniz" as guarantors, by charter dated 19 Jul 1074[511]. "Rodric Diaz et uxor mea Scemena" donated property to the abbey of Silos by charter dated 12 May 1076[512]. She attempted to retain control of Valencia after her husband's death but finally evacuated the city Apr/May 1102. The Anales Toledanos record the death in 1106 of “Dona Ximena su muger [del Cid]”[513]. m ([19 Jul] 1074) RODRIGO Díaz de Vivar "El Cid Campeador", son of Diego Laínez & his wife Teresa Rodríguez (Vivar near Burgos [1043]-Valencia [10] Jul 1099, bur Cardeña). Alférez of Sancho II King of Castile 1066. Lord and independent ruler of Valencia 1094-1099.

Using the naming patterns, Diego Lainez's father 'should' be named Lain, not Flain. Otherwise, he'd be Diego Flainez. Right? In fact, the Medlands page has several people with the surname Flainez, but this particular Diego who was the father of 'El Cid' is clearly Lainez and not Flainez.

So I'm sure the Medlands reference above has from Diego and Teresa to their son 'El Cid' and his wife Jimena Diaz correct, but I'm not sure Diego has the correct parents connected. I can't reconcile his Lainez surname with a father name Flain Nunez.