What qualifies as a mushroom?

All mushrooms are sporocarps, but not all sporocarps are mushrooms. So what's the distinction?

'Mushroom' is a culinary term, not a biological one

'Mushroom' is similar to 'vegetable' — both terms are defined by culinary people, not by biologists. [1]

Another answer is "it's complicated, since it's culturally defined".

Morphology

Wikipedia says "the word mushroom is most often applied to those fungi (Basidiomycota, Agaricomycetes) that have a stem (stipe), a cap (pileus), and gills (lamellae) on the underside of the cap". Mushrooms that fit this morphology are described as agaric.

Alternate morphologies:

• Bolete [📷]
• Truffle [📷]
• Puffball [📷]
• Stinkhorn [📷]
• Morels [📷]
• Cup fungi [📷]

Mushrooms that some people would argue about

Things that biologists wouldn't necessarily consider mushrooms (because they aren't agaric), but culinary people do, include:

• Morels [2] — Sac fungi, AKA ascomycete (and the sporocarp is known as an ascocarp)
• Truffles
• Polypores — Sheep polypore, turkey tail mushrooms, Berkeley’s polypore, dryad's saddle polypore, beefsteak fungus, hen of the woods / maitake, chicken of the woods, black-staining polypore, umbrella polypore, cauliflower mushroom, lingzhi / reishi, chaga, birch polypore, artist's conk.
• Tooth fungi — Lion's mane,