What is the definition of functions of Grassmann numbers? I understand there are some relevant questions, but none of them solves my issue.
From Atland and Simons (Condensed Matter Field Theory), the definition of functions of Grassmann numbers are defined by Taylor expansion. However, I do not understand the exact meaning of this definition. Let me take the function $f(x,y)=\exp(xy)$ as an example. I will use $x,y$ to denote ordinary numbers and $\alpha,\beta$ to denote the generators of Grassmann algebra.
1) If the definition of $f(\alpha,\beta)$ is: expand $f(x,y)$ as if $x,y$ are ordinary numbers, and then substitute $x\to\alpha$ and $y\to\beta$, then there would be ambiguities of how to order $\alpha$ and $\beta$. For example, $f(x,y)$ may be Taylor expanded to $1+xy+...$ or $1+yx+...$.
2) If the definition of $f(\alpha,\beta)$ is expanding $f(\alpha,\beta)$ directly with $\alpha$ and $\beta$ being Grassmann numbers, then one need to define derivative with respect to Grassmann numbers. However, the definition of derivatives are defined only for polynomials of Grassmann numbers and then for arbitrary function by Taylor expansion. Therefore, the definition of derivatives rely on the definition of functions.
Therefore, I have not been able to have a consistent picture of functions of Grassmann variable.
 A: *

*The main point when dealing with non-commutative objects [like Grassmann-odd numbers, which anticommute rather than commute], apart from, say, Taylor coefficients, we also need to specify an order of objects: The function/symbol itself does not amount to a full characterization.
A similar issue arises when we want to replace e.g. the classical Hamiltonian $H(x,p)$ with a Hamiltonian operator because $\hat{x}$ and $\hat{p}$ do not commute: We might also need a choice of operator ordering.


*Returning to one of OP's examples, if the quantity $f$ depends on 2 Grassmann-odd variables $\theta^1$ and $\theta^2$, it is customary to write $f(\theta^1,\theta^2)$. We know that there exist 4 (possibly supernumber-valued) coefficients $f_0,f_1,f_2,f_3$ such that $$f~=~f_0 + f_1\theta^1 + f_2\theta^2 +f_3\theta^1\theta^2.\tag{*}$$
Note that the order in each term of (*) is important. E.g the 2nd term $f_1\theta^1=(-1)^{|f_1|}\theta^1f_1$, where $|f_1|$ denotes the Grassmann parity of $f_1$. So if we chose another ordering convention, the above 4 coefficients may acquire sign factors.


*Another of OP's examples considers a function $f(\theta^1\theta^2)$. Here there are no ordering issues because the variable $x=\theta^1\theta^2$ is Grassmann-even, and therefore commutative, so that standard calculus applies [with the additional rule that the square $x^2=0$ is zero].
