I'd go with the first option and use a hole saw to cut a circle in the drain cover for the pipe (you want it fairly closely cut so that you don't get leaves finding their way in). You could replace it with one that's got a reinforced circular section ready for a cutout: https://www.toolstation.com/round-gully-grid/p71323
That bottom piece is really a joining piece. You should either get another of those pieces so you can direct the flow straight down, or your could replace it with a downpipe shoe: https://www.screwfix.com/p/floplast-round-downpipe-shoe-black-68mm/11456
I'd go with the first option and use a hole saw to cut a circle in the drain cover for the pipe (you want it fairly closely cut so that you don't get leaves finding their way in). You could replace it with one that's got a reinforced circular section ready for a cutout: https://www.toolstation.com/round-gully-grid/p71323