Off pavement is a good description, as opposed to off road.
I usually say gravel roads, and the jungle. Bush in NZ. Of course some places have impossible jungle areas, and some places have easy to ride off road areas.
We used to ride street bikes on gravel roads. But the bikes weighed a lot less. The GS1250 BMW was too heavy IMO, for the gravel; road to the Arctic. They crashed a lot. The faster bikes were a 300 Honda dirt bike, the KLR650 Kawasaki, a KTM500, and bikes like yours, the 650 twin Suzuki.
Many opted for larger bikes for the pavement part of the journey, then regretted it after crashing. A KTM 790 went well, ridden by a local cop. But he is not an average framed guy.
None of the bikes I have mentioned, would be good for serious off road riding, IMO.
Rather than an adventure, or fun, I would describe a tough cross country race of fifty miles, as the hardest thing I have ever done. It does get easier with experience. But it can be very hard on the body.