I kind of disagree with your ordering. For example, somebody who specializes in probability theory will typically know a lot of analysis and maybe other things. But according to your order, it is only 5th on the list.
On the other hand, somebody knowing manifolds, doesn't need to know anything about probability.

It would be better if you arranged the topics in some kind of tree. For example, the typical algebra branch will go as follows:

algebra -> linear algebra -> group theory -> commutative ring theory -> algebraic geometry

of course, I forgot to include Galois theory, semigroup theory, noncommutative algebra,... But I didn't include it because it doesn't fit nicely in the above branch. You will need more branches.

So there's no "highest" level of math. In fact, it is impossible to compare math knowledge of two persons. Unless they are in high school or undergraduates. But outside from that, two people may know a lot of different things...

Definitely taking higher level math courses. Mathematical proofs are very important in higher math courses. And proofs are very much a cultural type of thing that is best learned from experienced professors. It will be very beneficial to get accustomed to what is acceptable and what is not in the realm of proofs. I suppose you could try to teach yourself, but I think taking courses will be much more effective.

You can't really say that some subject is of a "higher level" than another. There are people who do research in linear algebra. Obviously this is not about the stuff you learn in a typical introduction to linear algebra.

You might say something like "high school level", "bachelors level", "masters level", "phd level", "research level" (while even those are not really well-defined, e.g. a first year Harvard student can have a 'higher level' than a masters student in Nigeria [sorry, just picking a random country]).

I agree that you can't really call anything "higher" than another; you can always look up something on wikipedia or take out a good math book on something relatively self-contained and read and "understand" it, but you may not have a proper appreciation of how everything is weaved together. Knowledge of mathematics is just a matter of reading from a paragraph to another, understanding of it is another thing

In terms of "highest level" (more prerequisites) for me so far would be:

A course: Calculus 1 (differential calculus)
Self-study (Where I feel like I have a thorough understanding of the subject): Linear algebra and basic abstract algebra

I don't plan in taking anymore formal math courses but I'm still very much interested to learn more math. I hope to learn stuff like topology and homological algebra, but it's a long way there. You could say I'm embarking on an independent mathematical expedition!

- Abstract Algebra
- Graph Theory and Combinatorics
- Set Theory
- Intro to Number Theory

As others have mentioned, ordering them is quite awkward because I don't believe you'd be able to find a logically consistent way of going about it. One might be able to order the level of study within a certain field/topic. For example, Linear Algebra from Axler or Shilov require a bit more mathematical maturity than a text like Strang or Lay. However, it would still be difficult to claim one of the texts is "higher" than the other.

High school Algebra
Precalculus
Univariate Calculus
Multivariate Calculus
Probability Theory
Linear Algebra I
Ordinary Differential Equations
Real Analysis
Complex Analysis
Algebra
Linear Algebra
Group Theory
Abstract Algebra
Numerical Analysis

Though on Abstract Algebra I only had 1 course, and that includes groups, rings, fields and boolean algebra altogether. And I'm still learning Numerical Analysis methods.

Next year I hope I'll be learning Topology, Differential Geometry and theory about EDOs and PDEs (only know some basics now), and hope to be learning stochastic calculus soon enough (for finance applications).