Nothing here triggers either requirement
Subdivision plat
Not triggered
Sec. 42-20(a) attaches to the subdivision of property. If you are not dividing property, it is not engaged — though a development plat may still be, below.
Development plat
Not triggered
Sec. 42-22 attaches to new construction or the enlargement of any exterior dimension of a building, structure or improvement. Without that, it is not engaged.
Class I, class II and class III are usually described as a hierarchy you are sorted into by the size of your subdivision. Sec. 42-23(a) says otherwise: class I and class II plats are optional and may be used in lieu of a class III plat. A class III plat stays available to anyone. What the criteria in Sec. 42-23(c) and (d) decide is which of the two lighter options you may choose — not which one you are assigned.
What genuinely forces a class III plat is narrow and listed in Sec. 42-23(b): a new street, a dedicated easement for public water, wastewater collection or storm sewer lines, or a vacating plat.
This is how the code reads; confirm with the city — or ask a builder who's actually pulled this permit in Houston. The authoritative text is the Houston Code of Ordinances; permitting questions go to Houston Permitting Center.