Round table

Therefore, if there was no damage (hit), then it is possible to retreat (run).

The so-called "hit" is the cast itself – it was performed (the hero cannot cast spells again in this round after using a spell).
To Wlk777 – don't take it literally that the rules must specifically use the conjunction "and"... simply "or" can combine identical concepts or separate different ones; therefore, everything after the comma applies only to non-combat spells.
What is combat magic? – it is the type of spell that, by definition, deals direct damage to units on the battlefield, so such magic does not need to be specified as magic

that in one way or another deals damage to the opponent.

Let's not argue unnecessarily, because if we nitpick to the end, then in theory, all spells that a hero can cast on the battlefield fall under the concept of combat magic; the remaining spells belong to the class of "adventure" spells.
So, literally, anyone who cast spells like haste, blessing, etc. in the first round, attacked the opponent and retreated, violating this rule.