Equality and equity sound a whole lot alike. They are only different by one low-density metallic element (yes, nerd joke), and people who aren’t paying close attention can be forgiven if they conclude they share much of the same space. Some would have us believe that equality needs only a few tweaks to generate the equity they assert our society lacks and needs.
They are, however, wholly incompatible principles, the equivalent of matter and antimatter. Where one exists, the other cannot.
Because we live in a time where equivocation and other language games are popular and accepted, definitions are necessary.
Equality means that a society applies one set of rules to everyone. What is allowed, restricted, or prohibited does not vary from individual to individual or from group to group. In equality, we can accept exceptions for minors or the mentally ill, since neither are capable of understanding and bearing responsibility for the consequences of their actions. That's the "adult of sound mind" qualifier, which is the limit of disparate treatment permissible under a premise of equality. Read it as "equal treatment under the law," if you wish - you get to the same state of societal organization.
Equity means that a society seeks to achieve equal outcomes for its members.
Since people are different intellectually and physically, they have different desires, motivations, and drive. A random two people with identical starting points will almost certainly follow different paths and achieve different outcomes.
This isn't a bad thing. In fact, it's a very good thing. This diversity of desires enriches us all in many ways, and is at the core of any thriving and growing society. Yes, it means that some people will make more money than others, but so what? No one has offered a good reason why wealth or income inequality in a society of equal treatment under the law is a bad thing. Those who assert it are making emotional plays on human instincts toward envy (and laziness and greed, for that matter), not rational arguments why such an outcome is undesirable. Do these same people argue that it was unfair for Aaron Judge to hit more home runs than everyone else last season?
Equality is mostly a "set it and forget it" sort of thing. When rules apply to everyone, they don't need people constantly judging, measuring, and coercing. Equity, on the other hand, is all about coercion. Since people are different, equal foundations will produce disparate results, and leveling those results requires constant force and perpetually unequal treatment. To pad up the "oppressed," and I'd double-bold the scare quotes if I could, everyone else's rights and liberties must be infringed.
This is why equity and socialism go hand in hand. Equity requires a coercive society, where one's freedom of action is restricted and burdened. Success is punished if it's achieved by the wrong sort of people, and failure is excused if it's achieved by the right sort of people.
Our society's founding philosophy is equality, but that doesn't mean we have achieved it. Apart from the obvious examples of slavery, segregation, and Jim Crow, there are countless examples of favoritism, cronyism, coercive redistribution, and other inequities. Take note of that word, "inequities," by the way. It's another trap. An inequity is an injustice. Taking from Jim to give to Joe is an inequity. Restricting Sally so that Sue may benefit is an inequity. Those actions are perversely pitched as "equity" when Joe and Sue are politically favored, and the excuse is that they are undoing the inequities generated by inequality. That's just word games again, and I reiterate that word games are a favored tactic of that crowd.
Achieving equity requires destroying equality. Achieving equality requires ignoring equity, and allowing people's outcomes to remain in their own control. Equality is a core component of a free society. Equity can only be achieved in a totalitarian, liberty-free society.
Equality has produced more improvements in living standards and human wealth - by far - than any other social structure yet conceived. Equity, or more accurately its pursuit, is a destroyer that has impoverished billions and killed tens of millions.
They cannot coexist. You can't have both. Choose wisely.
I preferred it when "equity" meant "the owner's stake" - as in, the equity you own in your home. Or referred to your ownership share of a business entity. Because it referred to what one had earned or built, through work or sacrifice. The modern use of the term stands the concept of earning on its head and treats wealth as something that just "is" and is to be distributed by government agency.
Another bit you hit on that is SO relevant was this: "Since people are different intellectually and physically, they have different desires, motivations, and drive. A random two people with identical starting points will almost certainly follow different paths and achieve different outcomes." That is TRUE diversity - not the coerced version based on skin color and sexuality that the left is foisting upon us. The only way the Left ever wins is redefining terms, and unfortunately, those opposed never stand against the definitional changes, which is why they lose - you can't win an argument if you concede to a redefinition of terms.