The piece of hardware in your last photo is called a shouldered lower link pin. Those are the pins used on tractor 3-point hitches, so a farm supply will have all sorts of them in different sizes. Something to think about is the these fit into a ball at the end of the lower link bars and that little hole is to hold a lynch pin, not really designed to see much force. I wouldn't trust it for safety connection if you bolt through the eye.
You can kinda see how they work here, lower left of the photo.
The first photo is what's called an eye nut. It's designed as a lifting point, so they are rated for a roughly similar use. If you back that with a grade 8 bolt with a thick washer it should be safe. Rating depends on size, but a 1/2" heavy duty steel carries a WLL of 3,600 lbs and a 5/8 a WLL of 5,800 lbs. Bear in mind that weight is along its axis with the bolt in tension, if you rotate the load to put the bolt in shear it might be different.
What might be better is a swiveling hoist eye, similar to the lifting eye nut but can move through a range of motion. Added complexity and cost, but the rating remains the same regardless of how you load it.
You could just use a standard shoulder eye bolt (or sometimes referred to as a turnbuckle eye bolt).
Or the various tie-down tabs.
How about bolting or welding on a swivel tab.