Correct: it doesn't exist

The best I can do, for the genera I'm more familiar with, is give you a list of species (below). This is based on the New Cactus Lexicon together with last year's Cactaceae Systematics Initiatives 25 which has always acted as a way of publishing new names for the NCL without cluttering up that book's pages. I've added some comments about some of the more confused names, provided I know something about it in the first place!
Austrocylindropuntiacylindrica
floccosa - most of the hairy plants you see (other than the easily recognisable A. vestita) are this, including a good proportion of those labelled rauhii, lagopa/lagopus. There are hairless forms, too.
lagopus - recently transferred (CSI25) to a new genus as Punotia lagopus. Many of the photos online are wrongly identified floccosa. malyanus is a synonym but photos of this are usually correct.
pachypus
shaferi - don't know much about this one
subulata ssp subulata - the one usually seen
subulata ssp exaltata
verschaffeltii - recently transferred (CSI25) back to Tephrocactus on the basis, I think, of molecular data.
vestita
Cumulopuntiaboliviana - split into 4 subspecies by the NCL: boliviana, dactylifera, echinacea and ignescens. I struggle to understand the differences.
chichensis
corotilla
ignota - I've never come across this plant for real and have seen very few photos, either.
iturbicola - new species (2011) published in CSI25. Like a small boliviana type?
leucophaea - a small-segmented (<3cm) version of sphaerica.
rossiana
sphaerica - a name that was used for lots of different things but has been slimmed down again by the removal of C. leucophaea, zehnderi, corotilla. Most of the things labelled as this are probably leucophaea, unless the segments are more than 4cm long.
subterranea ssp subterranea - previously in Maihueniopsis, Puna, etc. it's back here again!
subterranea ssp pulcherrima - first published as Tephrocactus pulcherrimus in 2000.
zehnderi
Maihueniopsisarchiconoidea
clavarioides
conoidea
darwinii
glochidiata new species (2011) published in CSI25. Like minuta.
glomerata - lots of things have been un-merged from this: darwinii, hickenii, molfinoi, ovata, platyacantha and there may be more to come.
hickenii - like a bigger version of darwinii??
minuta - mandragora probably belongs here.
molfinoi - plants labelled hypogaea belong here.
ovata
platyacantha
subterranea - recently transferred (CSI25) back to Cumulopuntia on the basis of seed and molecular data.
Punotia lagopus - A. lagopus renamed in 2011 (CSI25).
Tephrocactusalexanderi - includes geometricus
aoracanthus - includes articulatus v. ovatus and v. polyacanthus
articulatus - I've previously given my opinion on all the published varietal names
in this threadbonniae
halophilus - maybe! Vague species perhaps a form of aoracanthus or alexanderi.
molinensis - there are red and grey spined forms.
nigrispinus - an oddball in that it doesn't sit well in any genus, including Tephrocactus.
recurvatus - long known, recently described in 2011 (CSI25)
verschaffeltii - back from Austrocylindropuntia!
weberi - usually white spined but can be brown or somewhere between
Tunilla possibly we should be calling this genus
Airampoa - I'm not entirely sure. I share the opinion, expressed by others, that the species names in this genus are meaningless. I keep the names that came with my plants but don't hope to make any more sense of their classification any more.
Hope this is some use.
Tephroweb and
Opuntia del sur are both quite good.