Is there one 'correct' mindset for approaching mathematics?

The answer is yes and no. There is no single 'personality type' for a mathematician.

They can be visual, auditory, intuitive, or rigidly formal in their thinking.

However, there are certain core mental habits that are essential.

The habit of precision: Using definitions carefully and avoiding ambiguity.

The habit of skepticism: Questioning every step and demanding justification.

The habit of abstraction: The ability to zoom out and see the general structure.

The habit of persistence: The willingness to struggle with a problem for a long time.

So, while there are many valid ways to visualize or intuit a problem, the underlying thought process must be logical and rigorous.

You can be a 'visual thinker,' but you must be able to translate that visual intuition into a formal, logical argument.