Well, it occured that, while looking at the book, I found an indirect answer to my original question (and then some more...) in the most unexpected of places: The combat example of Grumpa!
As found in page 159, I'll copy it here:
Example: Our dear ogre is considered flanked, since
she is standing between two brigands – meaning that
the brigands have an Advantage against the big brute.
Grumpa solves this with a Movement Action, which triggers a Free Attack from each of her enemies (lucky that Grumpa confused one of them!). The ogre moves around the confused enemy, and receives a Free Attack from the other brigand. After making her move, the ogre has both the brigands in front of her, and is no longer flanked.
From the example, I can understand three things:
1) When being flanked, you can use a Movement Action to get out of the flanked position, but without needing to disengage from the melee combat. This Movement Action triggers Free Attacks.
2) Being engaged with two or more enemies doesn't automatically mean you are flanked; one of the opponents needs to take the Flanking Movement Action either when approaching you or during the melee combat.
3) If the Flanking Movement Action is done while already engaged in melee combat, then it triggers a Free Attack. [While this isn't specified by the example, it would be absurd to allow to retreat from flanking while remaining engaged, if the enemy could freely flank and attack you with advantage in his next initiative again.]
To point 3, although not specified anywhere, I'd add the option to use both the Movement and the Combat actions to Flank an opponent, and thus avoid the Free Attack at the cost of not being able to attack the flanked objective yourself. But this would allow your other companions to attack in their turns already with Advantage.
The way I see it, some additions or clarifications could be made to the RAW.
While I'm not very good at redacting rules, I'd propose something along these lines for the summary in page 163:
A. Movement Action
II. To flank an enemy already engaged
in melee with a combatant allied to
the flanking person. This gives both
flanking allies the Advantage against
their enemy. If already engaged in melee, the enemy gains a Free Attack.
IV. Disengage from melee combat:
To withdraw from melee allows
every engaged enemy combatant
to perform a Free Attack.
->> Could be changed to:
IV. Evasion in melee combat:
Every engaged enemy combatant gains a Free Attack and you may either withdraw entirely from melee or reposition yourself to avoid being flanked but remaining engaged.
Of course, the explanations in pages 161 (Special Actions - Disengage/Flanking) and 158 (the uses of a Movement Action) could be improved to include "Flanking an enemy while already engaged in melee (triggers Free Attack)" and "Maneuver to avoid being flanked (triggers Free Attack)".