Rule for l'

Cécile, I'm aware that the aspirate h exists, therefore I was careful to find reliable and reputable sources before posting, to not do this would do a grave disservice to my fellow learners.

It's l'Hérault according to the webpage of the departmental council of Hérault, at http://www.herault.fr/

The official tourism website uses l'Hérault: http://www.herault-tourisme.com/decouvrir-2-1.html

It is also l'Hérault on French wikipedia: https://fr.wikipedia.org/wiki/H%C3%A9rault_(d%C3%A9partement) which also uses l'Hérault for the river in the same area: https://fr.wikipedia.org/wiki/H%C3%A9rault_(fleuve)

French news uses l'Hérault: https://france3-regions.francetvinfo.fr/occitanie/herault/beziers/beziers-ligue-contre-cancer-herault-recoit-don-25-000-euros-1817750.html

The protected geographical indication (PGI/IGP) for the region's wine is «Pays d'Hérault»: https://www.wine-searcher.com/regions-igp+pays+d'herault

I'm not sure why you ignored the copious sources, including official local sources, that use «l'Hérault», in favour of a very generic site that is clearly going to lack local knowledge? Especially when there is a clear danger that anyone who is unfamiliar will incorrectly assume it  herald, «le Héraut».

Worst of all, the site you're relying on isn't even consistent about what it uses! Look at https://www.weekendesk.fr/idee-week-end-sejour/k2l/Languedoc-Roussillon-Bien-etre which says «Destination l'Aude, l'Hérault ou encore les Pyrénées Orientales pour profiter des bienfaits d'un week-end de pure détente.»

That's five examples, including the web site you're relying on, saying that you're wrong. Even if I explicitly search for «le Hérault» I get a preponderance of evidence that it's «l'Hérault».

Worst of all, the site you're relying on isn't even consistent about what it uses! Look at https://www.weekendesk.fr/idee-week-end-sejour/k2l/Languedoc-Roussillon-Bien-etre which says «Destination l'Aude, l'Hérault ou encore les Pyrénées Orientales pour profiter des bienfaits d'un week-end de pure détente.»

If you search hard enough you can find something to support anything. For example, I can find someone using «le Ontario» for the province. (I say the province, because «le Ontario» is used when it's used in the name of an organisation like «le Ontario Science Center» or a web address like «Visitez le Ontario.ca».) To be clear, that example for «le Ontario» turns up in my searches far earlier than weekendesk.fr does - I have not been able to get any results for Weekendesk at all, unless I explicitly include the name.

It would of course be silly to change it to «le Ontario» and ignore the vast number of examples to the contrary including ignoring the usage of the local government. However, that's exactly the precedent you've set with «le Hérault» where you've ignored the vast number of examples to the contrary including ignoring the usage of the local government.

Your dredging up of this obviously unreliable website is particularly galling because of how quickly 'French as in France' or a ruling of L'Académie Française is jumped on when it suits. (It's a different debate, but I think this attitude does a disservice to anyone who's trying to learn French for practical use, for example if they go to Brussels on business or encounters one of the words that's made it into mainstream french from Verlan.)

Finally, it's incredibly frustrating that the lesson still refers only to a vowel, despite your agreement that with a mute h it should be l'. The bottom line is that regardless of the example, the rule in the lesson is wrong, you know it's wrong, it's been a week, and it's not been corrected.