In both cases when mentioning the day of the week, they used an article, but the lesson states that when speaking of a specific day, one does not use

Hi Ann and Peter, 

In that particular case, talking of that romantic weekend, you have to use the article to convey -

On (the) Saturday you did this ... 

On (the) Sunday you did that ...

You would use , samedi, to talk of the particular saturday in relation to now .

Samedi, je vais aller voir ma mère à l'hopital =  On saturday I am going to go and visit my mother in hospital 

Hope this helps!

A bit trickier than that... If you were talking about the weekend just gone, you might say -

"samedi,  je suis allée visiter le musée d'Orsay à Paris"

To be clearer, you would use prochain et dernier but as it is still really close , people would understand your meaning.

In the meantime just to recap -

To say that you regularly do something on a certain day of the week, in French you would use

le  + day of the week:

Je fais mes courses le vendredi = I do my shopping on Fridays

Marie fait du yoga le lundi = Marie does yoga on Mondays

Je vais au bureau en vélo le mercredi = I cycle to the office on Wednesdays

But if you talk about something that is going to happen on one specific day, or has just happened in relation to today, you will use the day of the week only:

Anne va/est allée chez le dentiste mardi = Anne is going/went to the dentist's on Tuesday.

Son anniversaire est dimanche = Her birthday in on/this Sunday

J'ai rendez-vous chez le docteur mardi (prochain) = I have an appointment at the doctor's on Tuesday (or next Tuesday)

Hope this is a bit clearer!