Not really. It says the the Tuaren agrees to fight in the War in return for the Blood Debt they owed them for saving them. It’s specifically cited as the reason for fighting.
Then they return to Mulgore. NOT to the Horde.
Lets see.
Tauren weren’t a part of the Horde.
Rexar saves Baine.
Cairne and the Tauren Join the Horde in return.
Pretty darn straight forward.
Yes to the first part, no to the second.
They didn’t need to. Both went their own ways until the Horde needed help once again and sought out the Tauren again to seek their assistance once again.
Hypothetically, say if Baine wasn’t kidnapped when Rexar came to seek their help. Cairne wouldn’t have had any reason to offer up his people to fight against an enemy of the horde in a war that wasn’t theirs to fight. Unlike during the War of the Ancients, Cairne had no Blood Debt that he owed to the Horde so he was entirely within his rights to refuse to help the Horde, friendly relations or not.
You can argue that pass relations HELPED Cairne decide to join the Horde but you can’t HONESTLY say that he would have done so prior to the rescue of Baine.
No, that’s literally YOUR logic.
And the Answer would be no, that would not mean that the Tauren joined the Alliance. It means the Night Elves had helped the Tauren. That’s it.
What that would mean to their FUTURE relationship would depend on how those same events played out if the Night Elves did the same in each case.
The Lore would. The Tauren came to the Horde’s Aid because the Horde came to the Tauren’s aid once. That’s it.
In the same way the High Elves came to the Alliance’s Aid during Hyjal. Or the Shadowtooth Clan. Or the Barkskin furblogs. They all came to aid the Alliance but you can’t argue that they were members of the Alliance, right?