I never thought about using the logo only for the SEs, but that's an interesting idea. On the other hand, it would emphasize their "Special Edition-ness" as something distinct from the original product rather than a "restoration", which the makers of such SEs (whether in film or video games) are often at pains to de-emphasize, even when they involve hugely drastic changes to the original works.
Both with MI and the Star Wars movies the "Special Editions" have entirely replaced the original versions in the marketplace, for instance, except for the odd releases like the Limited Run Games MI box set and the laserdisc-transfer DVD versions of the SW Episodes 4-6.