The answer to the question ‘When is a Lily Not a Lily’ must be, when it does not belong to the Lilium genus of the Liliaceae family.
Despite the common names listed below bearing lily in the name they are not botanically Lilies.

Plants Masquerading as Lilies
Corn Lily
African ‘Aagapanthus’ lily
Arum lily
Belladonna lily
Bugle lily
Daylily ‘Hemorocallis’

Glory Lily
Guernsey Lily
Kaffir Lily
Lily of the valley
Lily Pink
Rain Lilies
Peruvian Lily

Plantain ‘hosta’ Lily
St Bernard or St Bruno Lily
Scarborough Lily
Toad Lily
Torch Lily
Voodoo Lily
Water Lily
Wood ‘trillium’ Lily

The real McCoy a group of Asian Lilies.

I was tempted to create this post after just ordering a crate of 200 Oriental Lilies to plant for cut flowers and a shock and awe feature in our late summer garden.

Of course most Aunt Lily’s are not from the lily family but there may be someone with a surname of Liliaceae and that would presumably make them qualify. Lonnie Donnegan had a dustbin full of lilies and was told to throw them away then but he couldn’t as ‘lilies still wearing them’.