With "not really under control" I mean "not completely under control". Sure, you are very right in the point, that Denmark temporarily conquered big parts of Pommerania, Mecklenburg, Holstein and Rügen. Last one at least for more than a couple of years.
But there are important arguments for the other view, too;
First point would be that the battles for this area lastet at least until 1214/1215, when emperor Friedrich II. officially acknowledged denmark´s supremacy in this area and transferred all vassal rights to the danish kingdom. After that, the occupation held for only 10 years. In the Battle of Mölln (1225) and Bornhöved (1227) Denmark lost almost all of these countries, except Rügen again. The conquered region were never under full and save control.
Second point would be that some countries like the south-west of the regarded region stayed fought area as long the occupation lastet. The Duchy of Schwerin was partially occupied but never neither acceptet the danish supremacy, nor was officially part of this danish province.
Third point; Another big part of the regarded region´s south was indeed part of the Duchy of Brandenburg - for example most parts of the former "Nordmark", which became as the so called "Mark Brandenburg" the core region of the Duchy of Brandenburg in the late 12th century. Brandenburg will even extend further north in later years and will for example include the whole region of Vorpommern (after the Battle of Bornhöved 1227, de facto 1231), so giving this faction to the Stralsund-Region is also very appropriate in view of its later development.
https://qph.ec.quoracdn.net/main-qim...d85c647954f7-c
If you want I could search for other sources, but here is at least a map showing the empire to the reign of Friedrich II. It implies what journey Brandenburg and the baltic coast will take and shows that Brandenburg is very fitting to this baltic region. The central part between the rivers Elbe and Oder is approximately the former Nordmark.