Replying to myself…
I have noted that page.id is only defined for posts. To make Cusdis comments work for all pages, it’s now changed like so:
<div id="cusdis_thread"
data-host="https://cusdis.com"
data-app-id="XXXXXXXXXXXXXXXXXXXX"
data-page-id="{{ page.path }}"
data-page-url="{{ site.url }}/{{ page.path }}"
data-page-title="{{ page.title }}"
></div>
<script async defer src="https://cusdis.com/js/cusdis.es.js"></script>