Exactly. What makes it index investing is low fees and simple, objective stock selection without human judgement. There are various well-studied criteria for value stocks.

An S&P500 index does stock selection too, it's just picking large-cap stocks instead of value stocks.

Large/Small are not correlated to value/growth. I for instance, invest pretty much all my money on ETFs such as ISCV, that invest on companies that fit the small and the value criteria. Over the decades, these two factors combined seemed to be the ones that on average gave the most returns.

I didn't claim they are correlated. I said some funds select based on value, and others based on size. Some do both.

> [...] objective stock selection without human judgement.

That's not completely true. The indices can involve plenty of human judgement, the S&P500 does, for example.

> Value investing is a strategy. Index funds are a vehicle.

Yes, that's correct. They are orthogonal concepts.