According to the FAA:
Transport airplanes are airplanes for which a type certificate is
applied for under part 21 in the transport category and that meet the
transport category airworthiness requirements. Multi-engine airplanes
with more than 19 seats or a maximum takeoff weight greater than
19,000 lbs must be certificated in the transport category.
In other words, there are two possible reasons why an aircraft is transport category: the manufacturer applied for a transport category type certificate, or the aircraft has more than 19 seats or MTOW greater than 19,000lbs.
The first reason refers to 14 CFR 21.21:
An applicant is entitled to a type certificate for an aircraft in the
normal, utility, acrobatic, commuter, or transport category, or for a
manned free balloon, special class of aircraft, or an aircraft engine
or propeller, if—
(b) The applicant submits the type design, test reports, and
computations necessary to show that the product to be certificated
meets the applicable airworthiness, aircraft noise, fuel venting, and
exhaust emission requirements of this subchapter
The "applicable airworthiness, aircraft noise, fuel venting, and exhaust emission requirements of this subchapter" for transport category aircraft are in part 25, AIRWORTHINESS STANDARDS: TRANSPORT CATEGORY AIRPLANES. If your aircraft meets those requirements and you get a type certificate from the FAA, then you have a transport category aircraft.
The second reason refers to 14 CFR 23.2005(a):
Certification in the normal category applies to airplanes with a
passenger-seating configuration of 19 or less and a maximum
certificated takeoff weight of 19,000 pounds or less.
That means, if your aircraft exceeds 19,000lbs then you can't certify it in the normal category (part 23), which leaves only transport (part 25).
By the way, the 23.3 regulation you quoted doesn't exist any more because part 23 has been completely rewritten.