Thanks @Achem and @porphyrin for your comments. I do not know if an Answer is the right format for this Comment but the comments do not allow enough character to develop my argument.
I think we are all in the same or close pages. My reservation is that we have to be a little nuanced when using this terminology, since otherwise it may leed to confusion.
I rewied this morning the the IUPAC's 3rd edition of Glossary of Terms Used in Photochemistry and look for the definitions of luminescence, fluorescence and phosphoresnce. I quote from the document:
Spontaneous emission of radiation from an electronically excited species or from a >vibrationally excited species not in thermal equilibrium with its environment.
Spontaneous emission of radiation (luminescence) from an excited molecular entity >with retention of spin multiplicity.
Phenomenologically, term used to describe long-lived luminescence. In mechanistic >photochemistry, the term designates luminescence involving change in spin >multiplicity, typically from triplet to singlet or vice versa.
Note: For example, the luminescence from a quartet state to a doublet state is also
Thus, both fluorescence and phosphorescence are examples of luminescence, independent of the spin of the states coupled in the transition.
Both triplet-triplet transitions (for example in O$_2$) and doublet-doublet transitions (for example in OH and NO) are examples of fluorescence. I am not familiar with any molecular transitions involving quartet or quintet states but that is my ignorance.
I guess that in some fields the molecular ground states are singlets and higher spin ground states and oddity, so I understand that the terms fluorescence, phosphorescence and luminescence can be used more loosely, even if it makes me unconfortable, but in that case I feel it is better to state in what context you are using the definition.