Should add that my budget is tiny and happy to use stones instead of slabs as that’s waaaay out of my diy skill set!

What is your question.

We have the same predicament and I am laying a path against the fence - it’s swampy too.
So will be creating a link between the top garden and bottom garden. I think if you make a patio, it will need to be quite wide and you will lose quite a bit of your grassed area

Looks like a fairly new build house? It’s quite common for lawns in new builds to suffer from compaction and thus create further problems with inadequate drainage. If the grass just feels a bit squelchy then I think some lawn aeration might be all that’s needed. Plenty of info online on how to do this but for a small section just driving a garden fork down as deep as it will go and giving it a wiggle every foot or so should help.

Patio....why not. Go for it! I did some slab laying last year (average diy..er) and whilst I wouldn’t attempt it on a large scale, for a small section its not so hard, back breaking work tho 😅. Do plenty of research, You Tube vids etc. and make sure you’ve got a spirit level 🙂.

A shrub or a small tree might also help with your drainage problem. Hydrangea might a good choice (the clue‘s in the name, they love to be hydrated.) Was thinking a couple either side of the step behind the fence might be great place for them to suck up the water. Depends if you wanted to lose more grass or not tho.

I think the problem with creating another hard area is that you garden will then have a tiny and almost inconsequential lawn, and will cause extra rain run off. The terrace area you already have near the house looks well constructed and quite large. I would agree that the lawn is probably suffering from being laid on poor compacted soil, and some general tlc would help terrifically. I would also get rid of the fence half way up the garden as it foreshortens the area making it look even smaller.

I think I would soften up your terrace with planters and your garden with some shrubs and climbers. Speaking from experience, having a seating area in full sun can be uncomfortable in the summer, and you just end up sitting under an umbrella anyway.

Make a zig zag or weaving path to make the space seem wider and at the end make a paved or deck triangular area to sit on. Planting at either side of path

It might be worth getting some professional advice/help with drainage, it might not just be compaction - there could be solid clay or stone below. Or at least dig a few deep holes to see what's under. Any planting may not thrive in badly drained soil.