The printer worked but the results, especially the precise ones with complex and circular borders did not look very well.
I decided to add a new part that would divert the belt a bit thus putting required tension on it.
After a short session with FreeCAD, even shorter slicing in new version of Slic3r and around half an hour of printing I had the necessary part. After adding the metal screws with some nuts and using spare bearing and washers the part was installed onto the printer.
The results after the upgrade were very satisfying. The circular objects looked circular :) After this moment I was convinced that was ready to work on parts that require decent level of precision.
A small digression- during the printing a major damage happened twice to the Y-axis pulley gear (you can see it in the picture above). It broke at its bottom. Of course I had to print the part and replace it. As you can image since the printer was broken it was not an easy task. Fortunately the glue held long enough to make the new print. It seemed that there will be no trouble but the part broke again. It seems the tension on it is too big (the engine's shaft is not going all the way through the pulley). The second time I made the part full in plastic- no saving for inner layers. I also treated the surface with acetone vapor to make it even stronger. Till now all is good. Of course I made an additional spare pulley (just in case) :)