The article states:
Limited Liability Company (...) files with the Texas Secretary of State. (...) This department keeps records of business names in a searchable database. Keep in mind you will still need to file a DBA within the county you are in.
Note the part about still needing to file a DBA with your county.
I am seeing conflicting information in 71.003(b) of the Texas Commerce Code:
This chapter does not require a (...) limited liability company (...) or its (...) members to file a certificate to conduct business or render a professional service in this state under the name of the entity as stated in the certificate of formation, application filed with the office of the secretary of state, or other comparable document of the entity.
This Dallas law firm says the same thing in their article What is a DBA and when do I need one?
Business owners who wish to conduct business under a name other than the trade name legally registered with the Secretary of State (SOS) during the formation process must submit a DBA (also known as an Assumed Name Certificate or Form 503) with the county clerk where their business is legally located.(Texas Business & Commerce Code [TBCC] § 71.103)."
Is the information on the houstontx.gov site correct?
If it is, what law or ordinance is it that requires the registration of a DBA for an LLC with the same name?