Because just calling them a Sea Priest is not enough. Again, where are they seen using the Light and/or the Void and are not corrupted? In WoW, a Priest is someone that uses one or both of those magic types. If there isn't any place where they are found using them, then they are priests only by title, but Shamans by their actual class.
Various Tidesage npcs are seen using spells like Lightning Bolt, Thundering Squall, Slicing Blast (being hit by a strong wind burst), and other water/wind/lightning based spells. Brother Pike (uncorrupted Tidesage) uses water to help heal a man (I forget his name) during the Stormsong questline. They are seen in Stormsong Valley summoning elementals with and without totems. The wands are just a means to bind them to their service. True, you don't need to call upon elementals to use elemental magic, but you do need to draw from the Elements (Which Shaman and Tidesages do). Otherwise Shamans couldn't do anything.
Arcane "Pure arcane appears white and/or violet. However, when used to bend other types of magic,
it takes on the characteristics and appearance of those magics. Examples of this can be seen with element-based abilities used by mages, or the bronze glow of chronomancy.
And why not? Tidesages are perfectly viable as Shamans. They use Shaman spells, have totems, work alongside water and air elementals (mages can't summon air elementals), they communicate with spirits and receive blessings from them. Outside of them sometimes being called Sea Priests and wearing cloth, they are just like Shamans. If we're going to go by slight differences, then Anduin is no longer a Priest since he wears plate and wields a sword, and Thrall is no longer a Shaman since he wears plate.