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The only other Water Type I could find that spawns naturally is Magikarp though, so as long as you stay clear of Water pools, Aqua Breed will be the only spawn getting a boost from the Water Lure.
For both it should also help to have the highest level on your team just under lvl 50. By default, spawn levels scale with your team's level, which can exclude some spawns that aren't available in those levels (uncaps at lvl 50+, at which point the whole range becomes available). That may not be enough to get rid of Ponyta (spawn levels 10-40), but it should eliminate the common first stage spawns like Ekans and Nidoran, passively increasing the changes for everything else because of that. Also Magikarp (lvl. 5-20), so you won't have to worry about it for Aqua Breed.
Last question- Would it help if my friend is using a Strong Fire Lure in addition to my Strong Fighting Lure? Or would the competition from the other fire pokemon just be too much.
Edit* Another question: Are certain Savanna biomes better? Do they need to spawn near grass?
No, all spawn events happen in relation to exactly one player, and only that player's boni will be applied to those spawns. If another nearby player doesn't have the same boosts as you, they'd only dilute your spawn pool, because half of the spawns will not receive your boni, but whatever theirs are. At most you can break even, if you both have the same.
The exact location can matter a decent bit. That does include the biome variant, yes - but also other aspect like viable Location Types (like Water, Underground, TreeTop, etc.; for Tauros you mainly want Land) exact time of day, sometimes how even the terrain is (for very large hitboxes), etc.
There are a lot of variables that play into that, spread across dozens of spawn sets. Spawns specific to certain biome variants are usually pretty rare though, so they shouldn't play much into the math there. If you have access to the "/checkspawns" command, you can check several locations for the highest chance of Tauros (the command doesn't differentiate between forms).