I guess for me it comes down to, how hard is it to get right settings manual for 560 iii vs 565 doing it from D610. If it'd easy to adjust and if one setting will work or be close for multiple pictures indoor the 560 may be perfect. If it's hard to adjust or a real pain, 565.
I don't anticipate using flash often as like I said, I really don't like it and don't want to always use flash on baby, it becomes annoying to be the guy constantly lighting up the room lol.
My notion is that flash is how we get the really good pictures. Light is the good stuff.
Not sure of your flash experience, but with a manual mode flash...
Flash Power Level depends on the flash to subject distance, for direct flash, the inverse square law. If a different distance or ISO or aperture or modifier like diffusion or umbrella, etc, then you set a different flash power level. Manually, you determine the power somehow (up to you), and set it.. For bounce, a different path and some unknown reflection coefficient, hard to predict (but TTL can meter any of this). And a handheld meter can meter manual flash too, but without one, it is trial and error.
With manual flash, you start by setting about the same power level on the flash that you used last time in a similar situation (distance, aperture, ISO, bounce or not, etc). Say maybe 1/16 power.
You take your picture, and then examine it on the camera rear LCD, and maybe the histogram. Probably it is too dark, or blown out too light.
You correct it by setting a different power level, a little more or less, depending on what you see it needed.
Then you repeat, until you get it right, like you want it.
Or, you could meter the flash, which is also what TTL does.
TTL is all that works while you are chasing kids around.
TTL flash, you just point and shoot. Then again, you examine results on rear LCD and maybe histogram, but it will generally be pretty close. Not always perfect, but pretty close. You can tweak it just a bit with Flash Compensation, and try again. With experience, we learn a few situations to know in advance what it will likely need.
If you are in a fixed studio situation, using umbrellas, etc, you probably want manual flash. There is no motion, your umbrellas cannot get up and follow motion. You get all your lights adjusted (normally using a hand held meter to meter each individually), and then they don't change, they stay like you set them.
You could use TTL (only for a couple of lights using Commander), but TTL is always metering, and can change a little, like metering a bit different when subject turns their head. But most prefer manual lights (and a hand held flash meter) in such fixed sessions.
But if you are walking around the room shooting many people, or for sure, if you are chasing fast kids around, you don't have any time, and you will want TTL to meter it for you. You check your first shot, and maybe adjust it slightly, but then the rest of the room is probably good to go, point and shoot flash. The TTL metering will track and keep up with normal little changes. TTL can be pretty wonderful - but sometimes it needs a little of our help too. At first, new users imagine the camera ought to do everything just perfect, and that attitude does not go very far either, not the real world. Photographers must learn to do what they see they need to do to help make the picture perfect. Mostly that is just paying attention, and fixing what they see needs it.
The Nikon top DSLR models (D7100, D610, D810, etc) are quite sophisticated in their flash features (Commander, etc). Just a few more dollars gets all of that use in a YN565EX, it can do whatever your camera can do. And it can do whatever a 560 can do - you can also do manual flash when you choose. A manual flash does not do any of that, it only flashes at the level you tell it to do.
Manual is just right in some fixed situations, an overwhelming choice. But it sucks when the subjects are moving around, and TTL can follow that. It is pretty nice to have the choice. Hope that helps. I hope the $30 difference does not cause you to miss many meals.