Minimal Sufficient Statistic of normal with known variance

Question

I have a problem in which $X_1, X_2, .., X_n$ follows $N(\theta,1)$ and I am required to compute the minimal sufficient statistic for $\theta$.

I can see from exponential family criterion, $T_1(x) = \bar{X}$ will be one sufficient statistic. Also, we can write $T_2(x) = (\bar{X}, S^2)$ as another sufficient statistic. Further, we have a a trivial sufficient statistic called ordered statistic as $T(X) = (X_{(1)},.., X_{(n)})$.

I know that, Any sufficient statistic which is a function of yet another sufficient statistic is called a minimal sufficient statistic. Now, I can view the statistic $T_2(X)$ as a function of $T_1(X)$. In fact, I can view $T_1(x)$ as a function of $T_1(x)$ only. So, I can say that both $T_1(X)$ and $T_2(X)$ can be viewed as a minimal sufficient statistic. Then, which one of these is best? I mean which one should i ideally report if asked about the minimal sufficient statistic. Further, we also know any one of one function of sufficient statistic is sufficient. I want to ask whether the function will be sufficient for the same parameter or transformed parameter. Because if it will be sufficient for the same parameter then we can generate more minimal sufficient statistics from here. So, my doubt is here only? Which one should be considered and reported when asked for this?

EDIT.1

So, this is the point from Casella and Berger Statistical Inference

It says:

Answer 1

Wrong statements:

1. Any sufficient statistic which is a function of yet another sufficient statistic is called a minimal sufficient statistic.

2. view the statistic $T_2(X)$ as a function of $T_1(X)$

3. any one of one function of sufficient statistic is sufficient

4. sufficient for the same parameter or transformed parameter

Corrected statements:

1. A sufficient statistic that is a function of all other sufficient statistics is a minimal sufficient statistic

2. there exists no function turning $\bar{x}_n$ into $(\bar{x}_n,s^2(x))$ (without $x$ being involved!)

3. a function of a sufficient statistic is not necessarily sufficient

4. sufficiency does not depend on the parameterisation of the distribution

---

The point in Casella and Berger is that here are two examples of sufficient statistics, hence with the same information about $\mu$, but one $T(\cdot)$ is summarising more than the other $T^\prime(\cdot)$ by eliminating $S^2$. (I am using Casella and Berger notations.) Nowehere is it mentioned that $T^\prime(\cdot)$ is a function of $T(\cdot)$.