I'm pretty sure he did in fact use Kurama's chakra in that instance, what makes you believe he didn't? He simply used it instantaniously in a moment of need, he didn't mold it manually. Remember when he fell down that cliff and summoned Gamabunta mid air to save himself? He obviously dipped into Kurama's chakra in order to do that, as he only was capable of summoning very small toads. The case is the same here, in a moment of need, he quickly dipped into Kurama's chakra and used it without signs of using it or molding it.
He was using his own chakra. Against Mizuki, he was pissed, & was able to tap into more of his chakra. Against Gaara, he was inspired by his comrades, & wanted to protect them no matter what, thus allowing him to tap into more of his chakra.
Back then, he could only use the Kyuubi's chakra whenever the Kyuubi decided to let him. Sometimes he asked for it, at other times the Kyuubi just gave it to him in a time of need. When he was falling down the cliff, it was both, he asked, & was in a time of need.
Additionally, Red chakra = Kyuubi's. There was no mention of Red chakra against Mizuki, he first used the Kyuubi's chakra against Haku.
After he finishes his combo he says something along the lines of "Lets go for 4 thousand!, this time i'll use both my hands and feet"
Then Gaara brings the shukaku out completely nuking all 99 clones.
It didn't passively leak until the land of waves arc and once the leaked chakra fuses, it permanently becomes part of his total reserves.
It's different from him requesting Chakra from Kurama himself.
In that situation he used all of his reserves, using Kurama's chakra to summon gamabunta. When Naruto uses the non fused Kurama chakra, his clones take on Kyuubi like features.
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