This 2020 Sage Adviceunofficial designer tweet from Jeremey Crawford directly answers this question, stating that "If you're a Storm Sorcery sorcerer and use Tempestuous Magic, the 10 feet of flight doesn't expend any of your normal movement that turn."
Any movement granted that doesn't specifically add to a speed type doesn't count towards expended speed if used. So, if a spell/ability says that you "move X feet", it is seperateseparate from movement speed, but if a spell/ability says you "gain a [type] speed of X feet", then it functions in coherence with the rule on page 190 of the PHB.