Chefmilitsa,
Just thought it might be helpful as you decide whether to refinish, a few things I discovered when I refinished my red '65...
It had been Dakota Red like yours and was refinished in the same color. Because Poplar is soft & porous, it absorbs paint or stain. When I stripped mine I saw the paint had imbedded deep in the grain. So even with all the surface paint gone, I could still see red in the grain. It took a bit more sanding along with several treatments with Formby's Refinisher (a very thin-watery and mild stripper) to get to pure wood.
But I only had to go this far because I'm going for a natural wood finish to match the headstock & neck. If you paint, the stripping won't have to be this in-depth.
Let us know what you decide to do and how it's goin.
Doug