The Penal Code, Chapter 46 Weapons, has a definition in a section about penalties for weapons offenses in a school zone. PC 46.11 (c)(2) says:
"School" means a private or public elementary or secondary school.
PC Chapter 28 has the same definition in a section on Graffiti.
In PC Chapter 71 Organized Crime, Section Sec. 71.028. GANG-FREE ZONES. (a) (1) says:
"Institution of higher education," "playground," "premises," "school," "video arcade facility," and "youth center" have the meanings assigned by Section 481.134, Health and Safety Code.
Going to Section 481.134 Drug Free Zones, it says:
"
School" means a private or public elementary or secondary school or a day-care center, as defined by Section 42.002, Human Resources Code.
Going to Human Resources Code 42.002, it does not define school, only "day-care center."
"Day-care center" means a child-care facility that provides care at a location other than the residence of the director, owner, or operator of the child-care facility for seven or more children under 14 years of age for less than 24 hours a day, but at least two hours a day, three or more days a week.
Those are all the formal definitions of "school" that I can find in the Penal Code. The PC does use language like "primary or secondary school," or sometimes "elementary or secondary school" to differentiate from "institutions of higher educations" (e.g. colleges and universities).
I'm sure the Education Code has something about defining "school" but searching that code is too daunting a task for me to undertake.
I did not see the work "kindergarten" or "kindergarden" anywhere, altho I didn't search specifically for it.
I hope this helps, but maybe not.
I'm sure there are a dozen threads on this already, but good luck searching them out.
ETA: I meant to mention I did not see any reference to "ballet schools" or "karate school" anything other than what I would take to be a "regular" elementary/primary or secondary/high school.