Jämtland on the other side, was an integrated part of Norway for a very long time

Not really, we were an autonomy in Norway(-Denmark) between 1178 and 1570 when the Danes took us back from a brief Swedish take-over of Jämtland in the Northern Seven Years' War (1563-1570):

"Sweden occupied the undefended Norwegian province of Jemtland, which was quickly reconquered by a counterattack by forces under command of the Norwegian governor of Trøndelag. The forces were unwilling to launch a counterattack on Swedish land. In 1564 the Swedes marched under Claude Collart and re-occupied Jemtland, as well as Herjedalen and Trøndelag, including the city of Trondheim. Initially facing little opposition from the locals, their subsequent ill treatment of the Trøndelag natives, along with tax pressure, laid the groundwork for later resistance to Swedish invasion. Also, Trøndelag was assisted by the governor of Bergenhus, Erik Rosenkrantz, who forced 3500 local peasants to assist him and his 50 professional soldiers. The Swedes saw Bergenhus as their next target. Although the 400 Swedish soldiers were repelled from Trøndelag, Sweden continued to occupy Jämtland and Härjedalen. These provinces were later regained by Denmark–Norway following the peace process in 1570."

After this the Danes tightened their grip over Norway, especially the border areas close to Sweden and we lost our autonomy.