The tritone being between final and fourth in lydian, to get a pure (minor) fourth down to final, or up from it, you lower the B to a B flat. The alternative would have been to avoid approaching them: which is so much easier in modes where the tones concerned involve neither final nor dominant. The first accidental was then a B flat. And a lydian (with its original final, not transposed) with a flattened B, is the scale of F major. The major mode transposed from modified "lydian" to pure naturals, called ionic by Glareanus, is of course between lydian and mixolydian:
He also instituted the eolic mode, between doric and phrygian:
Eolic looks like minor, right? Well, when melodic minor descends, it is simply eolic. Melodic ascending, as well as harmonic minors are not pure eolic. We are not quite there yet.