One reason for the popularity of equity arguments in the American political culture is that the 14th Amendment (enacted shortly after the U.S. Civil War) to the United States Constitution creates a binding, individually enforceable, legal obligation for governments to provide "equal protection" under the law to all persons. The legally enforceable dimension of this argument makes it a powerful one compared to norms that are not enshrined in the Constitution and hence are incapable over overcoming other laws to the contrary.
Equal protection arguments are usually framed as a mandate to treat similarly situated people equitably.
For example, in unequal school funding cases, both the equal protection clause, invoked, for example in the California case of Serrano v. Priest (1971) was brought as an equal protection case, while the Kansas case linked in the OP primarily relies on state constitutional guarantees of education but is also informed by equal protection requirements, and in terms of legal and political theory is drawing on the same body of thought about what it means to treat students equally.
I would dispute the characterization of the Kansas case as providing relief because "it was not equitable to school districts." It would be more accurate to say that "the Kansas education funding system was not treating school districts equitably," which does not really mean the same thing.
Equity as fairness in a one on one relationship between the two parties, while an older legal and political concept, if anything, is on the decline of an idea with philosophical currency and popularity relative to "equity" in the sense of treating people who are similarly situated the same. The former calls for an invocation of a natural rights means of political reasoning that is falling out of favor as legal positivism (i.e. the idea that legal obligations arise solely because an authoritative source decrees that the obligation exists rather than existing Platonically even in the absence of an authoritative imposition of that obligation) grows as a fundamental assumption of legal scholars, since equity in the sense of equality, rooted as it is in a broad enforceable legal standard with authoritative backing is a kind of idea that can be more powerfully advanced. "Equity" in the sense of fairness in a one on one relationship, in contrast, undermines the power of people with authority by having a separate source for binding obligations which people with authority have, naturally enough, disfavored.
For example, suppose that a school districts has funds left over at year end. An "equal protection" notion of "equity" might call for those funds to return to the state because if a school district has excess funds left over at year end, it doesn't need those funds as badly as schools that use up all of their funds and still can't meet their needs. But, a one on one relationship notion of "equity" associated with fairness might conclude that a district that is thrifty ought to be rewarded for their thrift and allowed to keep the funds left over at year end.
Desegregation cases like Brown v. Board of Education (1954) and its predecessors at the higher educational level were likewise formulate as cases about treating students equitably in education without regard to race (superseding the reasoning of Plessy v. Ferguson that separate but equal treatment was feasible as a practical matter, but giving lip service to the need to have equal treatment, even if it was separate, even then). Large bodies of equal protection law are described as "gender equity" and "language equity" cases.
Court litigation of constitutional rights, however, was very uncommon until the early 1900s and the equal protection clause was not widely litigated until the 1940s as a tactic of the early Civil Rights movement.
I'm not sure I could completely define equity, but it seems to me that
it is a notion of fairness that recognizes that each person (or
entity) has a different background. So it's different from equality (a
kind of fairness where everyone is given identical treatment) in that
it recognizes that in some cases fairness requires treating people
The notion of "equal protection" under the 14th Amendment has never meant a kind of fairness where everyone is given identical treatment, even when their circumstances are different in a relevant way. This concept of "equity" is more concerned with preventing improper distinctions being made than with substantively making treatment equal.
For example, no one under an "equal protection" version of "equity" analysis would ever suppose that school funding should be the same for all school districts regardless of how many students that they have, the number of students is a relevant difference.
On the other hand, "equity" in the sense of "equal outcomes" as opposed to "equal treatment given the relevant circumstances" has never had much currency as an implemented legal or political goal, although it is sometimes used rhetorically in contexts where the speaker is implying that the differences which are considered aren't relevant ones.