So I want to access details of the collection from within a page (or post) in that collection.

Going straight for site[page.collection] gives the list of docs in the collection the page is part of (page.collection just gives the collection label, i.e. how the collection is named in _config.yaml ).

You need to scan the list of all collections for the one whose label matches yours, then you can access its details.

{% assign collection = site.collections | where:"label", page.collection | first %}
{{ collection.description }}