Okay, I may not have the answer for you, but maybe I can help somehow...
As far as I understand, there are two different scenarios you are dealing with. One is an "Instant Hotspot" and the other is just a "WiFi Network".
Instant Hotspot is something new that Apple released in the last few months with iOS 8.1, I believe. Instant hotspot allows you to see your iPhone, enable personal hotspot on it and connect to it, all without touching the iPhone. It's pretty cool. In your image, the "BGP IP3" under the "Personal Hotspots" section tells you that it is enabled for this. Normal WiFi networks show up under "Choose a Network". You'll also notice that the Personal Hotspots that use Instant Hotspot also show signal strength and battery icons. The regular WiFi networks do not show this, as you can see.
I believe that at times, possibly after a failed instant hotspot attempt, your iPhone WiFi shows up in the regular "Choose a Network" setting. It's possible that this happens when you go to your iPhone's personal hotspot and enable it manually. Then you are able to connect manually to the WiFi on your iPhone, but not via the Instant Hotspot feature I'm referring to above.
So, I don't know exactly why the instant hotspot seems to be failing for you. I also don't know why the "manual hotspot" is requiring you to change the phone name in order to work between attempts. Here's what I do know:
1. Instant Hotspots are found when two devices have bluetooth enabled, see each other and are both logged into the same iCloud account. I take it that so far this part is working for you because you are seeing your iPhone as an Instant Hotspot option. What is not working for you is that when you try to connect, it fails, as shown in your image.
I had this problem, but from a Mac to my iPhone. I spoke with Apple Support and they noticed that I didn't have iCloud Keychain enabled in my iCloud preferences for both devices. Once I enabled it, everything worked. The engineer I spoke with thinks that the issue is because the iiPhone needs to share it's wireless passphrase with the device trying to connect to it so that it can be a truly hands off process.
I don't know that doing what I did and enabling iCloud Keychain on both devices will work for you. For what it's worth, I thought Instant Hotspot was only available for OS X and not iOS.
2. One other thing I learned from the Apple engineer is that the personal hotspot on the iPhone is only broadcasting it's WiFi network when a device is connected or when you go to the personal hotspot setting screen in the iOS preferences on the iPhone. So, if I got to the iPhone's personal hotspot preference and enable it, devices around me will be able to see my WiFi network. If I leave that screen, and no device has connected to it, then either immediately or within seconds the WiFi network will not be seen by other devices.
Of course if I connect to the WiFi while I'm on that screen, then I can leave the screen and it stays connected. But within a minute of disconnecting from the iPhones's WiFi, the personal hotspot will turn off until you go back to the screen. Note: The personal hotspot setting will still be enabled, but the broadcast will have stopped. Kind of annoying but probably a smart thing to do. It makes the instant hotspot option that much more desirable. Also, according to the guy I talked to at Apple, this is expected behavior though they don't really document it for us.
Maybe this also explains why leaving the personal hotspot screen on your iPhone to change the iPhone name, helps make the WiFi work again when you come back? Or perhaps changing the name triggers the WiFi without going back to the personal hotspot screen? Either way, if you can't get the Instant Hotspot to work and must do it manually, try leaving the personal hotspot setting screen on your iPhone and coming right back to it. See if that starts rebroadcasting the WiFi.
So maybe those two tidbits will help you? Does what I'm saying make sense?
Let me know if you have any questions. Let me know if anything I said ends up being helpful.