I would argue that quantum mechanics is actually not expressed in textbooks. Instead, they have exercises and sufficient hints. Try learning QM without doing any of the exercises and you will see what I mean. On the contrary, I bet you could learn QM just fine with just a few definitions and a series of well-designed problems.
"Starting with definitions" is the opposite of the language-less, intuitive way of understanding something. It means precisely that one has been provided with an effective formal language for the domain. The fact that you rederive some minor parts surely helps, but students don't routinely rediscover QM from scratch.
