Story has nothing to do with whether it's a reboot or not. Metal Gear Solid is a different game than Metal Gear. Almost entirely different. That's what's making it a reboot.
Except it does in fact play rather similarly(The MSX ones) and Metal Gear Solid is far from wildly different from its predecessors. Most of the changes in the game are due to the pushing of technology like voice acting and the limitations of the 2D perspective.
Otherwise from the perspective changes and the additions to complement that it's very much cut from the similar cloth from both a narrative and game play perspective. Yet with a 3D perspective you have to take many new factors into account and you have to make changes. As I already pointed out very few titles changed into 3D without fairly major revisions to make up for the new perspective they had to work with.
So you still haven't told me. If that's all it takes to make it a reboot why isn't stuff like Aliens which is nothing like the original a sequel? Obviously it's not right by your logic. It's completely different to the first and even has a different Director so it must be some kind of reboot, Metal Gear Solid has more in common with Metal Gear 2 than Alien does with Aliens.
Yet you seem to avoid that since they're fine counterpoints. Your qualifications for what makes a reboot don't make sense since several titles have broken what appear to be your rules for what makes a reboot. That or the Alien Quadrilogy is all just one reboot after another as it switches styles and decides to add numbers to the title and drop them on a dime and by what you've said about Metal Gear the Alien Quadrilogy has the same problems with being completely different to one another which must make it a series of reboots.
Being different doesn't make it a reboot like Resident Evil 4 for example, it is nothing like the titles that preceded it yet it's still a sequel to the events of RE3 and Code Veronica even if it's nothing like the previous Resident Evil games. Things can change, even rather drastically, while still not being a reboot.
I don't really see the logic behind what possibly makes it a reboot of Metal Gear, namely when the director himself stated his fairly well known(non-reboot) reasons for the name change.
Dino Crisis needs to rebooted. Controlling your character like you could in RE4/Dead Space would make a new game much, much more enjoyable. Give it a better story with more interesting characters, the obvious updates to graphics and audio, and better environments... Would be pretty awesome, I think.
The Syphon Filter series also needs to be rebooted (or even just brought back in general, but starting over seems to make more sense). I believe it's a first-party Sony title, so a release for the PS4 would make sense.
Vagrant Story is a game I would love to see remade, not necessarily rebooted. Keep the game mechanics (although fine tune them), and create a bigger, more interactive game world, tons of new enemies, voice acting, upgraded graphics and audio, new cutscenes, and more exploratory/platforming elements.