Okay I understand what you are saying now, but I believe that you are conflating two ideas here.

The first idea is about learning the concepts, and not just the specifics. There's a difference between memorizing a specific chemical reaction and understanding types of chemical reactions and using that to deduce what a specific chemical reaction would be given two substances. I would not call that intuition, however, as it's a matter of learning larger patterns, rules, or processes.

The second idea is about making things happen faster and less consciously. In essence, this is pattern recognition, but in practice it's a bit more complicated. Playing a piece over and over or shooting a basketball over and over is a rather unique process in that it involves muscle memory (or more accurately it involves specific areas of the brain devoted to motor cortex activation patterns working in sync with sensory systems such as proprioception). Knowing how to declare a variable or the order of operations, on the other hand, is pattern recognition within the context of a specific language or programming languages in general (as a reflection of currently circulating/used programming languages). I would consider both of these (muscle memory and pattern recognition) as aligned with the idea of intuition as you've defined it.

Rote learning is not necessary to understand concepts, but the amount of repetition needed to remember an idea after x period of time is going vary from person to person and how long after you expect someone to remember something. Pattern recognition and muscle memory, however, typically require a higher amount of repetition to sink in, but will also vary depending on person and time between learning and recall.