[Ethly]

It seems that the proper way to update unit information is OnAttributeChanged handler. Something like

function MyFrames_UnitTemplate_OnAttributeChanged(self, attributeName)
  if attributeName == "unit" then
    self.name:SetText(UnitName(self:GetAttribute("unit")))
  end
end

seems to work for me (I didn't test it in combat yet). I guess I overcomplicated with those secure snippets, there's no use for them at all for my purposes.