@GendoIkari
Correct, that's also how we play it, but I wanted to clearly point out that my opponent already had the flag before she started her turn.
@segura
The confusion is because of the Dutch wording in the rules book. It's saying the following for 'Flag', which is 'Vaandel' in the Dutch version:
"als je de Vaandeldrager pakt of vernietigt, neem je het Vaandel. Het Vaandel zorgt ervoor dat je bij het trekken van een nieuwe hand in de opschoonfase 1 extra kaart mag trekken. Dat geldt ook als je hand om een of andere reden uit minder dan 5 kaarten bestaat."
In Google translate it is:
"if you take or destroy the Banner Bearer, you take the Banner. The Banner allows you to draw 1 extra card when drawing a new hand in the clean-up phase. This also applies if for some reason your hand consists of less than 5 cards."
In the English version of Renaissance it is much more obvious:
"The Flag causes you to draw an extra card when drawing your hand in Clean-up. This is true even if your hand would normally be some amount other than 5 cards - for example if you played Outpost (from Seaside), instead of drawing 3 cards for your Outpost turn, you would draw 4."
So my opponent states that it not only activates when you're in the clean-up phase, but that it also applies when you have less than 5 cards (before you activated scholar).