Which test suit is better to say if PRNG is CSPRNG:TestU01 or NIST?
Please don't.

TestU01 or NIST?

Neither.  Trust what they say on the xoshiro256+ website. A CSPRNG is not defined by it's speed. It's defined by security against predictability, i.e. the next bit test. And that's determined by it's construction.
All that TestU01, NIST's STS and all the others do is check for uniform randomness of output (within a certain degree of confidence). A good PRNG will pass, as will a good CSPRNG. I don't think that a mathematical expression can be formed to enumerate the difference between an invertable PRNG and a non-invertable CSPRNG, other than something derived from $\text{xoshiro256+} \ne \text{CSPRNG}$. See note.
It's as mathematically defining the difference between apples and bananas.

My other question is that NIST statistical tes...

They say that because they are talking specifically from a computer security perspective. That's their raison d'etre. As in the last word of your linked title :-) NIST is not interested in PRNGs.

Note. With hindsight, something may be possible along those lines. Formulae can be written to invert PRNGs. How Brittle Are LCG-Cracking Techniques? discusses this with extensive formulae/code for a simple PRNG.  No complete formulae are (yet) possible for inversion of a CSPRNG. This is a form of mathematical distinction if you will.