Magic is Magic, and Magic is Feeling
Your Sorcerer, and other Magicky-Magic users, have powers that are inherently inconsistent. Whatever a piece of magic does at any given moment is different depending on the exact circumstances surrounding the user and their perception of the situation. The results are based on how they feel rather than any objective shared reality. Magicky-Magic users train their entire lives to gain mastery over their emotions and their perception of reality. Their study would be almost spiritual in nature, since they would essentially need to convince their inner consciousness that their upper consciousness should have control over their perception of reality.
Trained mages would be able to force themselves to feel whatever combination of ways to perform a particular specific spell, within some acceptable level of deviance. They have to immerse themselves in a feeling of true fiery rage in order to blast out a fireball, whereas an untrained mage might just light a match with a fake anger. This also explains where the "energy" comes from-- it temporarily saps the capacity for emotion and perception. A burnt-out mage would be "dead inside" and unaware, almost drunk, until they recharge.
In a way, this is still somewhat 'sciency', but it's under enough layers of emotion and feeling and abstractness that it doesn't feel like it at all.
Science is Science, and Science is Rules
On the other hand, our Sciencey-Magic friend has strict rules and specific physical actions required for certain spells. He has to follow exact formulae in order to make his 'magic' work. His magic is the equivalent of controlling a drone using lines of code. He has to provide the instructions ahead of time, and only then funnel power into his magic-machinery 'construction'.
Just like Science in real life, Sciencey-Magic progresses by experimentation to determine the rules he has to follow, documenting those rules, and iterating on those discoveries. The way he thinks about his magic, the way he uses it, would all be completely different from Magicky-Magic.
What's the Common Denominator?
The important part here, and the part that's most relevant to Writing, is that the way these are described, the words used, are completely different. The Sciencey-Magic uses words that are specific and measured, that describe strict rules. Magicky-Magic uses emotional words that evoke a flowery nature of perception-based reality.