IOS – Different UI for same version of YouTube official iOS app

It's not so much the name -- at least not the visible one, as an upgrade can change the visible name -- but it's the bundle identifier inside the app that identifies it and allows the OS to know it's supposed to perform an upgrade. The developer has to keep the bundle identifier the same in order to actually provide an upgrade and not a new app, so no, you're out of luck.*

*Note that there may well be some way to do such a thing if you jailbreak your device, but I have no firsthand knowledge of such.

Unfortunately, This is not possible due to current App Store limitations unless someone else with a jail broken iPhone that you trust has a copy of it that you can grab. One problem with dealing with the Apple App Store is that older versions are forever lost due to Apple's App Store polices. We should be giving concerned feedback to Apple to address this issue too, not just the app developers. So it looks like your best option if you need to have the Mint app, is to read up on the latest jailbreak, and insure your jailbreak path, then update your iPhone to the latest iOS, and then re-jailbreak it. The good news is that there is a untethered jailbreak for both iOS 5 and iOS 5.0.1 for the iPhone 4 as referenced in my comment above.