Both names are used and both make sense. You're right that it's the fourth scale degree that is sharpened, but as a tension, that scale degree is usually added to harmonies in the higher octaves, hence #11.
As for your question on the name "dominant" in phrygian dominant, it's about the major third. Phrygian has a minor third, and its basic seventh chord (made of the 1st, 3rd, 5th, and 7th scale degrees) is a minor seventh chord. If you change the third to a major third you get phrygian dominant, which has a dominant seventh chord as its basic seventh chord. That's also how this scale is used very often: over a dominant chord in minor. E.g., over A7(b9) resolving to Dm (6,7) you can play A phrygian dominant (which is just the fifth mode of D harmonic minor). Unlike the phrygian mode, phrygian dominant has the leading tone resolving to the tonic. E.g., in the key of the D minor, A phrygian dominant has a C# which resolves to the tonic D.
Any scale with scale degrees 1, 3, 5, and 7 that form a dominant seventh chord can be considered a "dominant" scale, e.g., mixolydian, mixolydian #11, phrygian dominant. Of course, there are also other scales that can be used over dominant chords, such as the half-whole (8-tone) scale, or the altered scale (seventh mode of melodic minor).