The Collaborative International Dictionary of English v.0.48:

Primate \Pri"mate\, n. [OE. primat, F. primat, L. primas, -atis
   one of the first, chief, fr. primus the first. See Prime,
   a.]
   1. The chief ecclesiastic in a national church; one who
      presides over other bishops in a province; an archbishop.
      [1913 Webster]

   2. (Zool.) One of the Primates.
      [1913 Webster]

WordNet (r) 3.0 (2006):

primate
    n 1: a senior clergyman and dignitary [syn: archpriest,
         hierarch, high priest, prelate, primate]
    2: any placental mammal of the order Primates; has good eyesight
       and flexible hands and feet

Moby Thesaurus II by Grady Ward, 1.0:

149 Moby Thesaurus words for "primate":
   Australanthropus, Australopithecus, Barbary ape, Bronze Age man,
   Cro-Magnon man, Galley Hill man, Gigantopithecus,
   Grand Penitentiary, Grimaldi man, Heidelberg man, Holy Father,
   Hominidae, Iron Age man, Neanderthal man, Oreopithecus,
   Paranthropus, Peking man, Pithecanthropus, Plesianthropus,
   Rhodesian man, Sinanthropus, Stone Age man, Swanscombe man,
   Zinjanthropus, aboriginal, aborigine, abuna, amphibian, ancient,
   angwantibo, antediluvian, anthropoid, anthropoid ape, antipope,
   ape, ape-man, aquatic, archbishop, archdeacon, archpriest,
   autochthon, aye-aye, baboon, back-number, behind the times, biped,
   bishop, bishop coadjutor, bushman, canine, cannibal, canon,
   capuchin, cardinal, cardinal bishop, cardinal deacon,
   cardinal priest, carnivore, cave dweller, caveman, chacma,
   chaplain, chimpanzee, coadjutor, cosmopolite, curate, dated, dean,
   diocesan, drill, ecclesiarch, entellus, eolithic man, exarch,
   feline, fossil man, gibbon, gnawer, gorilla, guenon, guereza,
   hanuman, has-been, herbivore, hierarch, high priest, hominid,
   humanoid, insectivore, invertebrate, langur, lemur, macaque,
   mammal, mammalian, man, man of old, mandrill, marmoset, marsupial,
   marsupialian, metropolitan, missing link, mountain gorilla,
   neolithic man, old hat, old-fashioned, old-timey, oldfangled,
   omnivore, orang, orangutan, out of fashion, out of season,
   out-of-date, outdated, outmoded, papa, patriarch, penitentiary,
   pontiff, pope, preadamite, prebendary, prehistoric man, prehuman,
   prelate, primitive, proboscis monkey, protohuman, quadruped,
   rector, reptile, rhesus, rodent, ruminant, rural dean, saki,
   scavenger, styleless, subdean, suffragan, troglodyte,
   unfashionable, ungulate, varmint, vermin, vertebrate, vicar



Bouvier's Law Dictionary, Revised 6th Ed (1856):

PRIMATE, eccl. law.. An archbishop who has jurisdiction over one or several
other metropolitans.



The Devil's Dictionary (1881-1906):

PRIMATE, n.  The head of a church, especially a State church supported
by involuntary contributions.  The Primate of England is the
Archbishop of Canterbury, an amiable old gentleman, who occupies
Lambeth Palace when living and Westminster Abbey when dead.  He is
commonly dead.