Can I try and help with a little history correction.

Historical fact: Decimus, not Brutus was the assassin who was Caesar's closest friend. The day before the assassination, he had supper at Caesar's house and persuaded Ceasar to ignore his wife's caution and go about business as usual the next day.

Now to the lesson:

It seems that the terms 'used on their own' and " you cannot use a conjugated verb" are interchangeable in the lesson.

A:le lendemain/la veille.... are used in a narrative strictly to give a sense of the chronology. You cannot use the terms as part of a 'temporal relative clause' to associate it with the event. If they appear in the same sentence as the described event, they are usually separated with a comma.

So: "Decimus, not Brutus, was Caesar's greatest  traitor'.  The day before he had dinner at Caesar's house.

    "Decimus, pas Brutus, était le plus grand traître de César. La veille, il dînait chez César."

      La veille is on it's own/not used with a conjugated verb in a relative clause.

B: le lendemain de+NOUN/LA VEILLE de+NOUN.. are used to overcome these restrictions when  you want to more syntactically and conceptually relate events to the chronology being referenced.

So: "The day before he assassinated Caesar, Decimus had eaten dinner at Caesar's house."

    "La veille de l'assassinat de César, Decimus avait dîné chez César." 

    See how important the chronological context is and how closely tied to the events in the 'telling' and how the   the syntax 'la veille + de + noun" is used to overcome the grammar restrictions.

==========================

LE JOUR D'APRÈS / LE JOUR D'AVANT and LE JOUR SUIVANT / LE JOUR PRÉCÉDENT are really 'weak' alternatives( according to the lesson)  to the VERSION of 'le lendemain and la veille' used in instance A!

But I think the lesson does them an INJUSTICE. Because, it seems, you NEED  these expressions when wishing to reference a specific DAY of the week in a past context.

EG: "le jeudi d'apres or le lundi suivant". It does not seem that you can say 'le jeudi lendemain' and you definitely shouldn't be able to say 'le jeudi veille'. So I would maybe add a line to the lesson, "when wishing to reference a particular day of the week in a past narrative, the expressions , "day of week.. D'APRÈS/SUIVANT or " day of week .D'AVANT/PRÉCÉDENT' is used instead of the generic le lendemain or la ville. 

Again the grammar restrictions 'on their own/not with conjugated verb' as illustrated in instace A applies.

And very conveniently:  

LE JOUR SUIVANT / LE JOUR PRÉCÉDANT + NOUN  = THE DAY FOLLOWING / PRECEDING [SOMETHING] is the way to overcome those grammar restrictions as was did in instance B with le lendemain/la veille.

I hope I am right and have helped you rather than steer you wrong. Because this is how I understand the lesson.

In the beginning of your reply you use the example, "La veille, il avait dîné chez César." This I understand. But could you say "la veille de dîner chez lui..." to talk about something that happened that day? (e.g. "La veille de dîner chez lui, il lui a parlé de la situation.")