Yurek,
thanks for the speedy action.
About the knots, if they work for you, do not change.
The biggest advantage of these knots is that you know how to tie them and that they have proved reliable for you.
The biggest disadvantage of the knot in the picture is that it may be hard to untie. It is based on a slipped thumbknot and fixed with another thumb knot and the thumb knot is known to be hard to untie.
The tautline hitch can slip, although it is rather secure if tightened properly.
I would advice people to use a locked sheepshank instead. The end of the sheepshank forms a kind of sheetbend when locked. It is hard to find a locked sheepshank on the net so I made my own picture. There are loads sites with more explanetions and more ways to tie the sheepshank. Hunt one down and tie the sheepshank that way. Next, you pull the ends of the rope / cord / string / sling through the loops at the end of the knot. You need to make a fairly big loop to get the sling through the opening but you can tidy the knot easily. When you want to undo this knot, you only have to undo this last tuck and with a little shaking the knot will fall apart.
Do not sling with an un-locked sheepshank. If un-locked the sheepshank will get undone when not under tension, always when you do not expect it.
For an easy made eye, there are many knots you can use. I would go for a non slip loop. The best known of those is the bowline but there are many more. Surf the web for one of the sites with knots and you find enough sites with practical knots
(But I know that you, Yurek, know all this.)
Willeke
http://www.slinging.org/images/vanDerHam/locked_sheepshank