What's with the last photo? Is that somewhere else in your yard, or an ideal design you are considering?

Please consider the following things that can't be changed: orientation to the southwest, access to the dog run, and drainage into this area.

• A southwest or western orientation means sun during the hottest time of the day. This can be a good thing in the winter, or if your climate doesn't get too hot during summers. If, your climate is hot, you will need shade. A tree may be the usual choice, but may not be the best due to the electric lines so close by. I don't see this area being best for sitting in.
• If the dog run is used, you can't block access to it. In fact, there should be a hardscape path to the gate to prevent the formation of packed dirt that becomes muddy when wet. Perhaps that would be a good use of your flagstone as long as it is thick enough not to crack when walked on.
• Drainage seems pretty well set; I'm not sure if the cutout in the fence is for a small dog, or to facilitate heavy runoff. I'm thinking that a rain garden might be the best use of this entire area, especially since there is also a downspout here as well.

A rain garden is not a water garden. The space collects water during heavy rain, but the water soaks in within a day. Locate a rain garden away from any foundation. Use plants that tolerate short inundations and also drier soil. A list of suitable plants depends on where you live. Do a search for rain garden plants for your state. Such lists are put out by garden groups, cities, and universities. A low spot is dug out (5-8" deep is sufficient) to collect water, and the extra soil creates a higher spot. You can plant in both the higher and lower levels. Turn the planting area and add compost to it. Gravel only needs to be located where water runs, such as from a downspout. The steeper the slope the more a channel should be lined with gravel and rock, and don't forget to leave an outlet for excess water from a downpour or extended rain.

Pay attention to the mature height and spread of plants, so tall plants don't block the view of smaller ones, and wide plants aren't placed too close to a fence, pathway or other plants. Initially the garden will be sparsely planted. Fill in with annuals until perennials and small shrubs take over.

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Last picture is the style of the yard I’ve been going for and is just to the left. I like the idea of incorporating rocks visually and for drainage. Well big rocks. I’ll hate the small rocks for the rest of my days after getting rid of dumpsters worth of it. Love your flower bed!

I very much love your garden style. I also love large rocks in my native landscape, though they are not without problems. There is no reason you can't use rocks in a rain garden. In the photo below the rocks slow the water as it flows from one point to another with the bottom group likely being the overflow point for heavy runoff.

French Broad Rain Garden - First Summer 2016 · More Info

I think it is important for rocks to be consistent in type and placement. So, you would want to use the same type and size of rocks in all your landscape. Rocks larger than 10x10x10 inches are, geologically speaking, boulders.

Here are some ideas: