Ph. yrs (1b) was a short detour after finding the good Nottingham University website. So back to the second group of place-names: those whose modern name bears some resemblance to Countisbury in containing the group -ntis-.
The first two had very similar names to Countisbury in Domesday. AD Mills’s derivation for these was the conjectural personal name *Centel. Domeday Chentesberie is not much different from the modern name – Kentisbury; as Contesberie is similar to Countisbury.
Nottingham picks up on Kantelesberi and Kantleber and suggests they include the British word for a district or border *canto-. This is the same element as is proposed for Quantock (Cantucuudu) and Cannington (Cantuctone, Candetone). There is also Early Welsh * kanto- meaning host, gathering: so, perhaps it was the fortified place where the war bands gathered, a meeting place?
Old Cornish also had cyntel/contel for a gathering together. However, more to the point, the attested early forms of Countisbury don’t include a Contelesberie, Cyntelesberie or Cantelesberie; so whatever the origin of Kentisbury it doesn’t help in suggesting where Countisbury came from. Another meeting place of British warbands up on Wind Hill would have been interesting, but there’s no evidence for that. No clue here, then.
The early forms of Trentishoe suggest trendel + hoh, a round hill-spur. Again, no early form of Countisbury like that.
So, still stuck on that first element, Cont-/Count-. Not Cunetus/Cynuit, not *Centel, not *canto-.
Ref: The medieval variant spellings come from the Historical Gazetteer of England’s Place-names.