Hi Nathan,
Yup, I'm afraid you are exactly correct as to what's happened; instead of separating the screen's frame from the phone frame, you've instead gotten between the screen itself and the frame that's supposed to be mounted on it.
I've run into this exact scenario on an iPhone X when I've been using some very cheap screens from AliExpress. The glue holding the display frame on isn't very strong, and definitely not as strong as that holding the screen onto the phone. So when you heat up the display, it's the cheap frame glue that gives way first, rather than the adhesive on the phone itself.
In the cases where that's happened to me, the screen itself either didn't survive or had obvious issues afterward. In your case, it's fairly safe to say you're now in the market for a replacement screen, sorry to say.
Here you can see a picture of the rear side of the display with the display frame intact. Compare that with the picture of yours, which is now missing the display frame - and can be seen still mounted on the phone itself.
Next steps for you:
- Order a replacement screen (sorry)
- Follow the screen replacement guide again, this time making sure to get between the display frame and the phone. You may be able to pry up from the inside to get it started since you obviously can't use a suction cup now. Go around the phone and separate the display frame; once that's done you can flip it back over onto the display and continue the screen replacement guide.
iPhone 12 mini Screen Replacement - iFixit Repair Guide
Update:
A couple more notes to add.
With two young grandchildren, both of whom have iPhone X phones, I have had to replace a LOT of screens; I have quite literally lost count of how many screens I've put on those phones. Fortunately, the iPhone X screens have dropped to rock bottom prices; I can buy OLED screens for about $15 now, where I thought I was getting a deal when I found them for $80 when I first started replacing them. The screens for your phone aren't going to be as cheap, but I did find some that weren't too bad in price. Here are some examples.
Amazon.com: TBUSKEJIE for iPhone 12 Mini OLED Screen Replacement Kit Full LCD Display 3D Touch Digitizer with Repair Tool for Model A2176, A2398, A2399, A2400 Black 5.4 Inch : Cell Phones & Accessories
AAA+++ OLED X Display For iPhone XR XS Max LCD Replacement For iPhone 11 12 Pro Max Display 13 14 Plus LCD 12 Mini 13 Mini 15
AAA OLED For iPhone X XR XS MAX Display 3D Touch Screen Digitizer For 11 Pro 12 Mini 13 Pro Max 14 Plus 15 LCD Replacement Parts - AliExpress 509
The Amazon screen is about $60 USD, while the AliExpress ones are $38 each. All three are OLED displays rather than the cheaper LCD replacements, although you can indeed spend less money if you want to go with the lower resolution, cheaper LCD parts. Be aware that your battery life will go down a bit since the LCDs have to add a backlight which takes more power to run.
Anyway, you'll note that the Amazon one says "LCD" in the title; despite that it is indeed an OLED screen; many vendors still use the term "LCD" to refer to a screen whether it's actually an OLED or not, so look carefully at any listing you come across to verify which it is. If the listing contains the word "Incell" then it is guaranteed to be an LCD not an OLED, since that's all they make.
To get the display frame off, you'll basically want to follow the screen replacement guide again, only this time make sure you're between the display frame and the phone itself. Since you won't be able to use a suction cup on the frame, you'll probably need to pry it up from the inside far enough to get a guitar pick in between; from there you can go around the phone slicing the adhesive.
Oh, just to be sure, you did remove the Pentalobe screws first, right? That's were a few of these problems have originated; from people trying to remove the screen without taking out the retaining screws, resulting in exactly this scenario. So if they're still there, be sure and take them out now before proceeding.
Good luck, and let us know how it all turns out!