Putting oracle forcing into context
$\sigma$closed  
$\Downarrow$  
uberoracleproper  $\Rightarrow$  strong $\bar{M}$proper  $\Longrightarrow$ $\Rightarrow$ $\bar{M}$proper 
proper  
$\Uparrow$  $\Uparrow$  $\Uparrow$  $\Uparrow$  
uberoraclecc  $\Rightarrow$  strong $\bar{M}$cc  $\Rightarrow$  $\bar{M}$cc  $\Rightarrow$  ccc 
$\Uparrow$  
$\sigma$centred 
Note: All arrows are strict. The only arrow which could possibly be added is $\sigma$centred $\Rightarrow \bar{M}$proper; this is unknown. The picture is supposed to show that strong oracleproper implies proper, but oracleproper does not imply proper.
Proofs of these implications and counterexamples to show strictness are given below (at least the ones that are not obvious, wellknown or mentioned in the previous post), working our way from left to right in the picture.

Proof Starting with a condition $p$ and $\delta \in \omega_1$, basically use the closure of the forcing to extend $p$ to a $(\delta, M_\delta)$generic condition.
$\Box$
To see that uberoraclecc implies uberoracleproper and that strong oraclecc implies strong oracleproper, one simply uses the fact that if a forcing is $\bar{M}$cc for a specific oracle $\bar{M}$ then it is $\bar{M}$proper for the same oracle (see Abraham’s notes for a proof of this latter fact).
The following claim (proof due to Martin is forthcoming) implies both that the notion of uberoracleproper is strictly stronger than oracleproper and that uberoraclecc is stronger than oraclecc.

The next claim can be found in Shelah 100.

Proof Assume $P = \omega_1$. Fix $p \in P$. Let $N$ be a countable elementary submodel of some large $H(\xi)$ such that $\bar{M}, P \in N$. Let $\delta^* = N \cap \omega_1$.
Let $S$ be the set of all $\delta < \omega_1$ such that there exists $q \leq p$ where $q$ is $(\delta, M_\delta)$generic which is club as $P$ is $\bar{M}$proper and $\bar{M}$ is a strong oracle. Therefore we may assume that that $\delta^* \in S$.
Now let $E \in N$ such that $E \subseteq P$ and dense in $P$. The set $C$ of $\delta \in \omega_1$ such that $E \cap P \upharpoonright \delta$ is dense in $P\upharpoonright \delta$ is club and $C \in N$ so $\delta^* \in C$. Also $E \cap P \upharpoonright \delta^* \in N$. We want to see that $E \cap P\upharpoonright \delta^* \in M_{\delta^*}$.
The set
$$\{N : N \cap [\omega_1]^{\aleph_0} \subseteq M_{(N \cap \omega_1)}\}$$
is club in $[H(\xi)]^{\aleph_0}$.
Why? Given $N_i : i < \omega$ in this set, we have
$$(\bigcup N_i) \cap [\omega_1]^{\aleph_0} \subseteq \bigcup_i M_{(N_i \cap \omega_1)} \subseteq M_{(\bigcup N_i \cap \omega_1)}$$
so it is closed. Thus the $N$ fixed above is a member of this set, which gives
$$E \cap P\upharpoonright \delta^* \in M_{\delta^*}.$$
Now by $\bar{M}$properness we have $E \cap P\upharpoonright \delta^* = E \cap N$ is predense in $P$ below $q$.
$\Box$
This next one though, proved by Martin, comes as somewhat a surprise and makes us think twice about bothering at all with the weak definition of oracleproper.

Proof Let $S$ be a stationary, costationary subset of $\omega_1$ and let $P$ be the forcing which collapses $\omega_1 \setminus S$. That is, conditions are continuous functions $f : \alpha \rightarrow \alpha$ where $\alpha$ is a successor ordinal and $f(\beta) \in S$ for all $\beta < \alpha$. This forcing is not proper, as models $N$ such that $N \cap \omega_1 \not\in S$ do not have $(N,P)$generic conditions.
Let $\bar{M}$ be defined as $\{M_\delta : \delta \in S\}$. Given $\delta \in S$ and $p \in M_\delta$ we will find $q \leq p$ which is $(\delta, M_\delta)$generic. Denote by $P \upharpoonright \delta = \{f \in P : f \subset \delta \times \delta\}$. Enumerate by $\langle A_n : n < \omega\rangle$ the set of $A \subseteq P\upharpoonright \delta$ such that $D \in M_\delta$ and are antichains in $P\upharpoonright \delta$. We may extend $p$ in $\omega$steps such that each $p_n$ forces that the generic intersects $A_n$ at a point $a_n$. Then $q = \bigcup p_n \cup \{(\delta, \delta)\}$ is a condition in $P$ and forces that for all $n < \omega$ the $P$generic filter meets $A_n$ at $a_n$ (i.e. is nonempty).
$\Box$
Finally we show that the oraclecc to oracleproper implications are strict.

Proof Let conditions in $P$ be finite partial functions $p : \omega_1 \rightarrow \omega_1$ which are weakly increasing. This is proper, but not Axiom A, see Jech Ch. 31 exercises.
To see that this forcing is oracle proper for any oracle, let $p \in P$ and $\delta$ be such that $P \upharpoonright \delta = \{p \upharpoonright (\delta \times \delta) : p \in P\}$ (happens on a club). Then for $M_\delta$ we let $q = p \cup \{\delta, \delta\}$ which is $(\delta, M_\delta)$generic.
$\Box$