I noticed that Firefox and Epiphany (a Mozilla-derived browser) were downloading the same video upon page reload.
Looking at Firefox’s cache, I could not locate a single video file. However, I noticed that during download, the video was in cache.
The problem was that the default maximum cache to store is set to 50 Mb, which Firefox respects.
As the YouTube videos I was watching were big (~30 Mb), they were not kept in cache. The solution was to increase the maximum cache size.