“Saturation” is a generic term for regions that are useless for an amplifier (i. e., are strongly non-linear on \$I_{\mathrm B}\$ or have β less than 1), but are far from cut-off. Most authors don’t apply due efforts to formulate a usable definition. There exists at least one English book not negligent on the matter:
E. Ramshaw (Google Books info) or
R. S. Ramshaw (as written on the cover)
Power Electronics Semiconductor Switches
Springer Science & Business Media, 2013
The book specifies a distinct quasi-saturation region, that is non-linear (meets “4.”) but the collector current still has a healthy direction. When \$V_{\mathrm BC}\$ passes through zero (≈ “2.”), then a hard saturation mode develops, that meets the definition “1.”. Ī̲’d call it “on–on region”.
Also, Ī̲’d characterize the region immediately preceding the hard saturation as “low-voltage”, having low (but positive) β, see comments and my answer in Why does the collector current direction remain the same in saturation and active region? for details.
What about the en.Wikipedia definition (as of today) of regions? It can be traced to various sources where authors were unwilling to get into details of these marginal modes. Quasi-saturation is close to the hard saturation region (albeit operates in a different way), and many authors were satisfied with the simplistic picture presented in @jonk’s answer. What about the stuff from es.Wikipedia under “3.”? It’s just incompetent rubbish.