Is there an operational definition, or a published study where a
general definition is given, which would allow me to recognize in a
generic molecule (or at least an organic molecule) whether a given
bond is rotatable or not?
It depends on your purpose. Why do you want to identify some bonds and call the rotatable? For instance, PubChem, one of the biggest databases of molecules, defines rotatable bond as follows:
Rotatable bond is defined as any single non-ring bond, bounded to
nonterminal heavy (i.e., non-hydrogen) atom.
To some extent all this boils down to the following IUPAC definition of free rotation:
free rotation (hindered rotation, restricted rotation)
In a stereochemical context the rotation about a bond is called 'free'
when the rotational barrier is so low that different conformations are
not perceptible as different chemical species on the time scale of the
experiment. The inhibition of rotation of groups about a bond due to
the presence of a sufficiently large rotational barrier to make the
phenomenon observable on the time scale of the experiment is termed
hindered rotation or restricted rotation.
At usual lab conditions a rotation around a single non-ring bond, bounded to nonterminal heavy atom will be indeed almost free, thus, PubChem definition can be used as a very simple rule-of-thumb to identify such bonds. But in general it depends on conditions. And purposes.
With respect to protein backbones being rotatable or not: I'm not a speacialist in this area, but Wikipedia article on peptide bond claims that (emphasis mine)
Significant delocalisation of the lone pair of electrons on the
nitrogen atom gives the group a partial double bond character. The
partial double bond renders the amide group planar, occurring in
either the cis or trans isomers.
So, this partial double bond character might contribute in protein backbones being not quite rotatable. Besides, the Wikipedia article also claims that (emphasis again mine):
In the unfolded state of proteins, the peptide groups are free to
isomerize and adopt both isomers; however, in the folded state, only a
single isomer is adopted at each position (with rare exceptions).
So, at least in the unfolded state protein backbones are rotatable. The fact that only two configurations (cis and trans) are possible is irrelevant. The notion of rotatable bond does not imply having few specific possible configurations.