The name of the category is trill. The R sound in the linked demonstrations of Latin is not a "guttural" sound but an alveolar trill, [r], where the tip of the tongue vibrates against the roof of the mouth, which is traditionally believed to have been the R sound in Latin, though arguments have been made that it was more likely an alveolar tap, [ɾ], the sound in American English atom.
The trill in French, or in modern Paris/Brussels/Montreal/etc French at least, is uvular rather than alveolar, which may be categorized under "guttural". In the uvular trill, [ʀ], it is not the tongue that vibrates. Instead it's the uvula, the piece of flesh dangling at the back end of the roof of the mouth. In many modern varieties of French, the tongue does not make as narrow a constriction as a typical uvular trill, so that it is better classified as a fricative [ʁ] or approximant [ʁ̞], though some trilling of the uvula can still be heard even in such cases.