(a) The public security authorization that authorizes a fully registrable public security shall designate the registrar for the security.
(b) The registrar may be:
(1) the comptroller;
(2) a home-rule municipality with a population of more than 100,000, as to a security of the municipality;
(3) a county with a population of more than 100,000, as to a security of the county;
(4) a bank, including a commercial bank, at which the principal of the security is payable; or
(5) a trust company organized under a law of this state.
(c) Designation of a county as registrar for a public security issued by the county is effective only if the commissioners court of the county makes the designation with regard to an issuance of debt.
(d) The county treasurer or the county officer who has the powers and duties of the county treasurer shall perform the registration duties for a county that is designated a registrar.