A notarization is a notarization and it has a specific legal effect. Any valid notarization will do. The notary must be acting in their geographic jurisdiction and must have a currently valid notary commission at the time, but that is all, although, if the particular notary has a conflict of interest and notarized anyway, the business might be justified in disregarding it, and in some circumstances with indicia of fraud or duress or undue influence, the business might be justified in a delay to investigate the situation.
Colorado has adopted the Uniform Power of Attorney Act which at Colorado Revised Statutes ยง 15-14-720 allows someone with a duly notarized power of attorney to go to court to force a firm or government agency to recognize it and imposes liability on firms that fail to recognize it. Texas has also adopted this Act, so an almost verbatim identical provision should be present somewhere in the statutes of the State of Texas.

The circumstances under which a power of attorney may be dishonored under the Act (in addition to POAs drafted in a foreign language that must be translated) in Texas should be substantially similar to this Colorado list:
(2) A person is not required to accept an acknowledged power of
attorney if:
(a) The person is not otherwise required to engage in a transaction
with the principal in the same circumstances, including, without
limitation, the circumstances set forth in paragraphs (a.3) and (a.5)
of this subsection (2);
(a.3) The agent seeks to establish a customer relationship under the
power of attorney and the principal is not currently a customer;
(a.5) The agent seeks services under the power of attorney that the
person does not offer;
(b) Engaging in a transaction with the agent or the principal in the
same circumstances or acceptance of the power of attorney in the same
circumstances would be inconsistent with any federal or state law,
rule, or regulation other than as set forth in this part 7;
(c) The person has actual knowledge of the termination of the agent's
authority or of the power of attorney before exercise of the power;
(d) A request for a certification, a translation, or an opinion of
counsel under section 15-14-719(4) is refused;
(e) The person in good faith believes that the power is not valid or
that the agent does not have the authority to perform the act
requested, whether or not a certification, a translation, or an
opinion of counsel under section 15-14-719(4) has been requested or
provided;
(f) The person makes, or has actual knowledge that another person has
made, a report to a governmental agency having authority to protect
the welfare of the principal stating a good faith belief that the
principal may be subject to physical or financial abuse, neglect,
exploitation, or abandonment by the agent or a person acting for or
with the agent; or
(f.5) The person has an apprehension, formed in good faith, that the
agent or person acting for or with the agent has acted or is acting,
in any capacity, either unlawfully or not in good faith in dealing
with the person and the person is investigating in good faith to
determine whether the person may, based on the results of the
investigation, form a good faith belief that the principal may be
subject to financial abuse, neglect, exploitation, or abandonment by
the agent or a person acting for or with the agent.
The parallel Texas statutory section states:
Sec. 751.206. GROUNDS FOR REFUSING ACCEPTANCE. A person is not
required to accept a durable power of attorney under this subchapter
if:
(1) the person would not otherwise be required to engage in a
transaction with the principal under the same circumstances, including
a circumstance in which the agent seeks to:
(A) establish a customer relationship with the person under the power
of attorney when the principal is not already a customer of the person
or expand an existing customer relationship with the person under the
power of attorney; or
(B) acquire a product or service under the power of attorney that the
person does not offer;
(2) the person's engaging in the transaction with the agent or with
the principal under the same circumstances would be inconsistent with:
(A) another law of this state or a federal statute, rule, or
regulation;
(B) a request from a law enforcement agency; or
(C) a policy adopted by the person in good faith that is necessary to
comply with another law of this state or a federal statute, rule,
regulation, regulatory directive, guidance, or executive order
applicable to the person;
(3) the person would not engage in a similar transaction with the
agent because the person or an affiliate of the person:
(A) has filed a suspicious activity report as described by 31 U.S.C.
Section 5318(g) with respect to the principal or agent;
(B) believes in good faith that the principal or agent has a prior
criminal history involving financial crimes; or
(C) has had a previous, unsatisfactory business relationship with the
agent due to or resulting in:
(i) material loss to the person;
(ii) financial mismanagement by the agent;
(iii) litigation between the person and the agent alleging
substantial damages; or
(iv) multiple nuisance lawsuits filed by the agent;
(4) the person has actual knowledge of the termination of the agent's
authority or of the power of attorney before an agent's exercise of
authority under the power of attorney;
(5) the agent refuses to comply with a request for a certification,
opinion of counsel, or translation under Section 751.201 or, if the
agent complies with one or more of those requests, the requestor in
good faith is unable to determine the validity of the power of
attorney or the agent's authority to act under the power of attorney
because the certification, opinion, or translation is incorrect,
incomplete, unclear, limited, qualified, or otherwise deficient in a
manner that makes the certification, opinion, or translation
ineffective for its intended purpose, as determined in good faith by
the requestor;
(6) regardless of whether an agent's certification, opinion of
counsel, or translation has been requested or received by the person
under this subchapter, the person believes in good faith that:
(A) the power of attorney is not valid;
(B) the agent does not have the authority to act as attempted; or
(C) the performance of the requested act would violate the terms of:
(i) a business entity's governing documents; or
(ii) an agreement affecting a business entity, including how the
entity's business is conducted;
(7) the person commenced, or has actual knowledge that another person
commenced, a judicial proceeding to construe the power of attorney or
review the agent's conduct and that proceeding is pending;
(8) the person commenced, or has actual knowledge that another person
commenced, a judicial proceeding for which a final determination was
made that found:
(A) the power of attorney invalid with respect to a purpose for which
the power of attorney is being presented for acceptance; or
(B) the agent lacked the authority to act in the same manner in which
the agent is attempting to act under the power of attorney;
(9) the person makes, has made, or has actual knowledge that another
person has made a report to a law enforcement agency or other federal
or state agency, including the Department of Family and Protective
Services, stating a good faith belief that the principal may be
subject to physical or financial abuse, neglect, exploitation, or
abandonment by the agent or a person acting with or on behalf of the
agent;
(10) the person receives conflicting instructions or communications
with regard to a matter from co-agents acting under the same power of
attorney or from agents acting under different powers of attorney
signed by the same principal or another adult acting for the principal
as authorized by Section 751.0021, provided that the person may refuse
to accept the power of attorney only with respect to that matter; or
(11) the person is not required to accept the durable power of
attorney by the law of the jurisdiction that applies in determining
the power of attorney's meaning and effect, or the powers conferred
under the durable power of attorney that the agent is attempting to
exercise are not included within the scope of activities to which the
law of that jurisdiction applies.
None of those conditions concern a failure to use a particular designated notary.
USPS Form 1583 is a form presented to the U.S. Postal Service, so that only person that could honor or decline to honor it, would be the U.S. Postal Service, which does not, in fact, require that any particular notary be used.
Some businesses require something different than a notarization called a "guaranteed signature" rather than a notarization for certain transactions. A notary is not necessarily authorized to guarantee a signature (which is a private sector parallel system to notarization with greater liability for the signature verifying party who has been certified to meet the system's standards and to be solvent if a signature is erroneously guaranteed). It is possible that a notarized power of attorney would not be accepted by a business for a transaction because a guaranteed signature, rather than a notarization is what is actually required.