Awesome, you're way ahead of me on both counts!! Those chairs you ordered might be the best of both, half swing, half egg!
And you've got your conduit in, brilliant! You mentioned two lights for $1600. Where would he put them? You have a lot of stairs. You asked about "
stair flagstone tops." Let's get our vocabulary straight, so we're talking about the same thing. A stair has two parts, a tread and a riser. The riser is the vertical section, the tread is what you walk on. And do you still have the three planters? It's hard to tell from the pics and new drawings. Looking at the pool, standing on the stairs, lets call the planter nearest the kitchen #1, then the one by the stairs #2 and the one up where the lounge chairs are #3. Are you talking about mounting the lights flush, in the risers, just under the treads? That would illuminate two treads really well, then cast some shadows on the others, leaving them much darker. That would work, as long as they all had some light on them, but not ideal. You'd really need one for each riser to make it even, but even then the top tread would be dark. I was thinking of one of these (and they have lots of other styles):

They're adjustable, and throw big circles of light, up to 16' I think. You put one or more in each of the planters, 1,2 and 3 (free standing, not embedded in the stone work) and the whole area around them will be lit, along with the plants, too. The light in planter #1 will light up that whole lower area along with some of the lower steps. The light in planter #3 would light that whole top area including the top tread, along with the plants in planter #3 and some of #2. One or two lights in planter #2 will illuminate the steps and plants in both #2 and #3, along with some of the area in front of planter #1. Even some of the pool, depending on the plants. You with me so far? The rub, though, might be how close you can get the lights in planter #2 to the stairway. They'll be back a bit, and the retaining wall next to the stairs, the one that forms planter #2, will cast a shadow on part of the stairs. See what I mean? The stairs will still be lit a bit by the light thrown from planter #1, but maybe not enough. So the half of the stairway next to the lawn will be well lit, the other half not as well.
That's where the other types of lights come in. Look at the fourth one from the left, above. That could be in planter #2, hanging out over the wall a bit, and really light up the stairs. Or you could put one of those in planter #1 and aim it just so, right at the stairs and really light them up. But because of it's "salami cut" shade, you can adjust those just so the light hits the stairs, but not someone's eyes as they come down the stairs. Just to make things a bit more confusing, these lights will cast shadows, so for example, that spot light I just describe, in planter #1, would cast a shadow of the person walking up the stairs onto the stairs. Not ideal, but it'll still be diffused so it's not like you'll lose sight of the stairs. So there are trade offs. Nothings going to be better than one of these in each riser:
Or maybe some number less in the wall along the stairs? But I don't think that would work well for the top two steps. Those lights would throw less light elsewhere, only on the stairs, and maybe that's all you want. Or you could combine the two effects, even on different circuits, so you can turn on some subtle stair lights, or turn on the whole show and light up all the plants and trees and stairs in your yard. Depends on what you're after (which, I haven't forgotten, is just some stair lights, before I got carried away!).
If you were going to plant shrubs and bushes and ground cover next to the stairs, I'd say put a couple of those path lights near the stairs and you'll have a beautiful effect on the plants and you'll have very well-lit stairs. But if that's to be lawn, as per your rendering, then you don't want to be mowing around light poles. You could have a little strip of plants, next to the stairs, then lawn beyond that. That would look nice and be a place for the path lights, which would also illuminate the lawn at night. The plants would visually soften the transition between the stairs and the lawn. Personally, that's how I'd do it. In fact, I'd ditch that lawn all together. Here in California, lawns are so 1950s! It's all about the water now a days. I'd plant some nice ground cover and some shrubs and bushes and turn that whole area into a little garden showcase, all lit up at night. Less water, no mowing!
I don't know if I'm being helpful, I just know how much I love my yard lights, and how versatile all the various styles are. And how much more you could light up for as little as $500. For $1600 you could do your whole yard and it would be amazing at night!