You've pretty much just described what spin-offs are :/ A starting point that also acts as a continuation of something that preceded it. Metal Gear Solid is definitely a series, and it plays into a much larger Metal Gear Series, but it's still a spin-off of the original.
A comparison with Nolan's Batman trilogy doesn't quite work since Metal Gear established a use of regular sequential naming conventions (with numbers and all). A spin-off takes a point from a series and literally spins off into its own tangent. Whether or not that tangent is a one-off or creates its own is just up to them - I again point you to Ghost Babel, which jumps off from the exact same point as Metal Gear Solid does. Ghost Babel is a direct sequel, too, by your definition, but then now we have a discrepancy - two sequels to the MSX series. It's actually a little silly to just state that the Metal Gear series as a whole is Metal Gear (MSX) to Metal Gear Solid 4 when we have Ghost Babel sitting there right alongside the Solid series.
Ghost Babel and Solid suppose two different divergences from MSX series. One is just infinitely more popular and did more to help the franchise skyrocket into fame.
Also - the originals Metal Gear and Metal Gear 2: Solid Snake were on the MSX computer system, not the NES. The NES version of Metal Gear was a butchered abomination