I have some pedantic confusions when it comes to spin structures. Let $B$ be a nice space (like a CW complex) and let $\xi$ be an oriented vector bundle on $\xi$. If I choose a metric on $\xi$, then I have the corresponding frame bundle $F_{SO}(\xi)$. Then I can talk about spin structures on $F_{SO}(\xi)$ being $Spin$-principal bundles $P$ together with an equivariant map $f: P \to F_{SO}(\xi)$. Further, I can say when two spin structures $(P_1, f_1)$ and $(P_2,f_2)$ are equivalent, namely that there exists an equivariant map $P_1 \to P_2$ commuting with the maps $f_1$ and $f_2$.
Now, if I don't choose a metric, I can't talk about $F_{SO}(\xi)$ (although I still have $F_{GL^+}(\xi)$ for example). So I have a few related questions:
(1) Can I define a spin structure without choosing a metric?
(2) Suppose I choose two different metrics, say $g_1$ and $g_2$ on $\xi$, and then I form the two different associated frame bundles (I'll be pedantic with notation for a moment) $F_{SO}(\xi, g_1)$ and $F_{SO}(\xi,g_2)$. Suppose also that I have chosen spin structures $f_1: P_1 \to F_{SO}(\xi, g_1)$ and $f_2: P_2 \to F_{SO}(\xi, g_2)$, what does it mean for these two spin structures to be equivalent?
I'm confused because I usually see the set of equivalence classes of spin structures on $\xi$ referred to with no mention of the metric. An equivalent definition of spin structures is that they are elements $\sigma \in H^1(F_{SO}(\xi, g); \mathbb{Z}/2\mathbb{Z})$ that (for $n > 1$) restrict nontrivially to the fiber. In a similar spirit to (2) (and in fact this would give an answer to (2)):
(3) For different choices of metrics $g_1$ and $g_2$ on $\xi$, is there a canonical bijection between $H^1(F_{SO}(\xi, g_1); \mathbb{Z}/2\mathbb{Z})$ and $H^1(F_{SO}(\xi, g_2); \mathbb{Z}/2\mathbb{Z})$ that preserves the spin structures?
Thanks for the help, I know these are pedantic and probably not so interesting questions... but I'm confused.