I would bet this is caused by a tire with rubber that got "old". This tends to happen to MTB tires that are repeatedly ridden on mud, and I had some tires with this behaviour before.
The problem seems to be insuficcient friction between the rubber and the rim around the bead, specially on those tires with a material resembling fabric on that region (like it seems to be your case, judging by the first photo), and are worse on the front tire since it applies traction only while braking - the rotational force is always applied in the same direction.
One measure to take, as already mentioned, is to put a higher pressure on the tire. It might allow you to get home without damaging a spare tube, but it's not the ideal solution since the slipping tendency is still there anyway, and sooner or later the tire ends up slipping, most probably when you don't expect.
One other measure, that actually works very fine, is to "reactivate" the rubber on the groove where the rim and the tire "seat" against each other, considering that:
- In tires with old rubber, many times the said surface is "shiny", or "glassy", or "vitrified", or "mirror-like";
- In tires with fabric in the bead, the rubber "disappears" with use, and the slippy underlying fabric is left to make contact with the rim.
Both conditions contribute for the slipping due to reduced friction coefficient. So, to REACTIVATE the grippiness of the material, you should:
- Take the tire from the bike and wash it with plenty of water until it is completely wet. Since the tire is off, you can wash the rim either;
- Brush the bead groove vigorously with a properly-sized brush, applying some abrasive soap mixed with water. Mechanics soap could be great but maybe too violent. I had excelent results with toothpaste (seriously!), and even with glycerin soap without abrasives. In any case, adding water enough to get a good foam is much better than just rubbing the dry product on the rubber. Rinse repeatedly after this with pure water, aiding with the brush;
- Optionally, at last you can apply those products to make car tires "shiny". These tend to somehow protect the rubber from getting dry and glassy again.
- Important: assemble and inflate the tire while it is still wet.
I had tires almost unusable because of this slipping problem, and the problem disappeared like magic after performing these steps.
As an additional fact, any modifications I tried to make in the rim (cleaning, sanding, filing, grooving) had no effect.
Hope this helps!