Relationship between Spin(3), SU(2), unit quaternions, and SO(3)

There may be some short-hand / informal statements that are tripping me up, but I am getting confused trying to understand the relationship between Spin(3), SU(2), SO(3), and the unit quaternions.

Trying to find information online, many discussions seem to say SO(3) and SU(2) are isomorphic (for example wikipedia). Mathworld says SU(2) is isomorphic to $O^+_3(2)$ which I'm not quite sure how that relates to SO(3) (I have not seen that notation before). While others state SU(2) is isomorphic to the unit quaternions which are in turn a double cover of SO(3). Which seems to suggest there is a lot of "short-hand" discussion going on and sometimes people say isomorphic ignoring a double cover? Or maybe I just misunderstand, and a double cover doesn't really matter for some reason?

My best effort of trying to figure out what is going on seems to suggest:

$$Spin(3) \cong SU(2) \cong \{q \in \mathbb{H} | q\bar{q}=1 \} \not \cong SO(3)$$

Is that close, or are even more of those actually double covers?
What is the correct relationship between these groups? (and what does $O^+_3(2)$ denote?)

$Spin(3), SU(2)$, and the unit quaternions $Sp(1)$ are all isomorphic; this Lie group is also sometimes referred to simply as its underlying manifold $S^3$. $SO(3)$ is diffeomorphic to $\mathbb{RP}^3$ and so is not diffeomorphic to $S^3$, although its double cover is $Spin(3)$ (and hence also $SU(2)$ and $Sp(1)$).

One possible source of confusion is that all of the corresponding Lie algebras are isomorphic, and some sources (especially from physics) do not closely distinguish between Lie groups and their Lie algebras.