The US embassy in Delhi is huge. I wouldn’t be at all surprised if a small specops team were staying in the barracks there. A counter-terror team might only need to be 3 or 4 guys if that, so it’s not like there’s a huge US army base in India somewhere. If this were the case, both the US and India would be correct, since the US is saying there is a counter-terror team in India (at the US embassy), but India is saying there is no terror team in India (on Indian soil, since the US embassy is US territory).
The US Embassy is technically American soil in India. Likewise, the Indian Embassy in Washington, D.C.
American, Australian and Indian troops have regularly held exercises in the varied terrains of all three countries.
Alaska-based exercises mimic Indian Himalayan exercises.
Jungle warfare schools of the Indian military in India’s far-east has been attended by American exchange forces in the past, and will continue to do so, in the future.
So on, and so forth...