I wouldn't re-use this strainer. I've never seen one designed like this before. I think the idea is the notch you highlighted slips up over the protrusion that I highlighted:

then you rotate the ring so it rests on the protrusion, and tighten the thumb screws. In your case, I think, the thumb screws were over-tightened and the ring forced down over the protrusion. That's why it was so hard to remove. Once you loosened the thumb screws it should have easily rotataed and just fallen off. It is quite likely to be damaged and will not hold well any more. IF you tighten the screws enough to hold the gasket well, the ring will start riding down over the protrusions and you'll get back into this stuck-askew configuration. Replace it.

Here is a different design where the plate bearing the thumb screws is held on by the drain threads rather than by little notches. This tends to work well.

I wouldn't re-use this strainer. I've never seen one designed like this before. I think the idea is the notch you highlighted slips up over the protrusion that I highlighted:

then you rotate the ring so it rests on the protrusion, and tighten the thumb screws. In your case, I think, the thumb screws were over-tightened and the ring forced down over the protrusion. That's why it was so hard to remove. Once you loosened the thumb screws it should have easily rotataed and just fallen off. It is quite likely to be damaged and will not hold well any more. IF you tighten the screws enough to hold the gasket well, the ring will start riding down over the protrusions and you'll get back into this stuck-askew configuration. Replace it.

Here is a different design where the plate bearing the thumb screws is held on by the drain threads rather than by little notches. This tends to work well.

If you think your old one is higher quality, keep the parts and if the new one does start rusting you can maybe mix and match the parts. The old basket and ring with the new thumb screws and their triangular plate.

I wouldn't re-use this strainer. I've never seen one designed like this before. I think the idea is the notch you highlighted slips up over the protrusion that I highlighted:

then you rotate the ring so it rests on the protrusion, and tighten the thumb screws. In your case, I think, the thumb screws were over-tightened and the ring forced down over the protrusion. That's why it was so hard to remove. Once you loosened the thumb screws it should have easily rotataed and just fallen off. It is quite likely to be damaged and will not hold well any more. IF you tighten the screws enough to hold the gasket well, the ring will start riding down over the protrusions and you'll get back into this stuck-askew configuration. Replace it.

I wouldn't re-use this strainer. I've never seen one designed like this before. I think the idea is the notch you highlighted slips up over the protrusion that I highlighted:

then you rotate the ring so it rests on the protrusion, and tighten the thumb screws. In your case, I think, the thumb screws were over-tightened and the ring forced down over the protrusion. That's why it was so hard to remove. Once you loosened the thumb screws it should have easily rotataed and just fallen off. It is quite likely to be damaged and will not hold well any more. IF you tighten the screws enough to hold the gasket well, the ring will start riding down over the protrusions and you'll get back into this stuck-askew configuration. Replace it.

Here is a different design where the plate bearing the thumb screws is held on by the drain threads rather than by little notches. This tends to work well.