Is there a name of this particular Buddha, and what is the mudra called, what does it mean, and what is in his left hand and what does it mean?
That fits the description of the "the Medicine Buddha",
Bhaiṣajyaguru is typically depicted seated, wearing the three robes of a Buddhist monk, holding a lapis-colored jar of medicine nectar in his left hand and the right hand resting on his right knee, holding the stem of the Aruna fruit or Myrobalan between thumb and forefinger.
... but I have no better explanation than that.
I suppose the "medicine nectar" is 'the deathless' e.g. as in this answer -- see also 'nectar' is mentioned in other topics on this site.