The stalk of the structure sheaf on the étale site at a geometric point is the strict henselization of the corresponding local ring, cf. Section 4 of Milne's lecture notes on étale cohomology, or EGA IV.

There are two things to say regarding the commutation issue raised in anon's comment (which you can view as a continuity condition if you want):

• in the case of Prop 6.12 of the lecture notes on motivic cohomology, $F=\mathbb{Z}_{\operatorname{tr}}(T)$ for $T$ a scheme of finite type. Because of the finite type assumption, every correspondence $\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}}\to T$ is in fact represented by a correspondence $U\to T$ where $U$ is an étale neighbourhood of $x$ in $X$; therefore the stalk of $\mathbb{Z}_{\operatorname{tr}}(T)$ at $x$ can be computed as $\mathbb{Z}_{\operatorname{tr}}(T)(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$. (See also Exercise 1.13 in the lecture notes.)

• in the argument for the rigidity theorem, the functors that are being considered (i.e. presheaves with transfers) are defined for schemes of finite type - in some sense, there is actually no definition of $F(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$. So the functors are extended to local rings (or more general, essentially finite type schemes) by the definition $F(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})=\operatorname{colim}F(U)$ where $U$ runs over all étale neighbourhoods of $x$ in $X$. The necessary continuity problem is defined away... (As a side note: this is not possible in the original application of the rigidity theorem to K-theory because K-theory is defined for all rings. Fortunately, K-theory commutes with directed colimits as in the definition of the local rings, so the continuity requirement is again met.)

Your statement is correct, in both cases, the exactness of the sequence is established by looking at the stalks. Strictly henselian local rings show up because the stalk of the structure sheaf on the étale site at a geometric point is the strict henselization of the corresponding local ring, cf. Section 4 of Milne's lecture notes on étale cohomology, or EGA IV.

To identify stalks of the sheaf with sections over the local ring, you need some condition - the limit needs to commute with the functor as mentioned in anon's comment. (This can be seen as a continuity condition). In both cases mentioned in the question, continuity works:

• in the case of Prop 6.12 of the lecture notes on motivic cohomology, $F=\mathbb{Z}_{\operatorname{tr}}(T)$ for $T$ a scheme of finite type. Because of the finite type assumption, every correspondence $\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}}\to T$ is in fact represented by a correspondence $U\to T$ where $U$ is an étale neighbourhood of $x$ in $X$; therefore the stalk of $\mathbb{Z}_{\operatorname{tr}}(T)$ at $x$ can be computed as $\mathbb{Z}_{\operatorname{tr}}(T)(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$. (See also Exercise 1.13 in the lecture notes.)

• in the argument for the rigidity theorem, the functors that are being considered (i.e. presheaves with transfers) live on a category of schemes of finite type - in some sense, there is actually no definition of $F(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$. So the functors are extended to local rings (or more general, essentially finite type schemes) by the definition $F(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})=\operatorname{colim}F(U)$ where $U$ runs over all étale neighbourhoods of $x$ in $X$. The necessary continuity problem is defined away... (As a side note: this is not possible in the original application of the rigidity theorem to K-theory because K-theory is defined for all rings. Fortunately, K-theory commutes with directed colimits as in the definition of the local rings, so the continuity requirement is again met.)

The stalk of the structure sheaf on the étale site at a geometric point is the strict henselization of the corresponding local ring, cf. Section 4 of Milne's lecture notes on étale cohomology, or EGA IV. Regarding the commutation issue raised in anon's comment: in the case at hand (Prop 6.12 of the lecture notes on motivic cohomology), $F=\mathbb{Z}_{\operatorname{tr}}(T)$ for $T$ a scheme of finite type. Because of the finite type assumption, every correspondence $\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}}\to T$ is in fact represented by a correspondence $U\to T$ where $U$ is an étale neighbourhood of $x$ in $X$; therefore the stalk of $\mathbb{Z}_{\operatorname{tr}}(T)$ at $x$ can be computed as $\mathbb{Z}_{\operatorname{tr}}(T)(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$.

The stalk of the structure sheaf on the étale site at a geometric point is the strict henselization of the corresponding local ring, cf. Section 4 of Milne's lecture notes on étale cohomology, or EGA IV.

There are two things to say regarding the commutation issue raised in anon's comment (which you can view as a continuity condition if you want):

• in the case of Prop 6.12 of the lecture notes on motivic cohomology, $F=\mathbb{Z}_{\operatorname{tr}}(T)$ for $T$ a scheme of finite type. Because of the finite type assumption, every correspondence $\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}}\to T$ is in fact represented by a correspondence $U\to T$ where $U$ is an étale neighbourhood of $x$ in $X$; therefore the stalk of $\mathbb{Z}_{\operatorname{tr}}(T)$ at $x$ can be computed as $\mathbb{Z}_{\operatorname{tr}}(T)(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$. (See also Exercise 1.13 in the lecture notes.)

• in the argument for the rigidity theorem, the functors that are being considered (i.e. presheaves with transfers) are defined for schemes of finite type - in some sense, there is actually no definition of $F(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$. So the functors are extended to local rings (or more general, essentially finite type schemes) by the definition $F(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})=\operatorname{colim}F(U)$ where $U$ runs over all étale neighbourhoods of $x$ in $X$. The necessary continuity problem is defined away... (As a side note: this is not possible in the original application of the rigidity theorem to K-theory because K-theory is defined for all rings. Fortunately, K-theory commutes with directed colimits as in the definition of the local rings, so the continuity requirement is again met.)