What is the shape of large cardinal tree in implication strength order? There are two natural orders on large cardinal axioms.
(a) Consistency strength order
$\sigma \leq_C \theta \Longleftrightarrow ZFC\vdash Con(ZFC+\theta)\longrightarrow Con(ZFC+\sigma)$
(b) Implication strength order
$\sigma \leq_I \theta \Longleftrightarrow ZFC\vdash \theta\longrightarrow \sigma$
It is well-known that these are not same and many large cardinals (e.g. Woodin cardinals) have different positions in large cardinal tree when we endow it with different orderings.

Question 1. Are there any other important orderings on the tree of large cardinal axioms?
Question 2. What is the shape of large cardinal tree in implication strength order? Is there any diagram somewhere in the texts which summarizes the results on direct implication of large cardinal axioms?

I hope somebody give me an explicit diagram which shows the differences between large cardinal tree in implication and consistency strength orders or at least based on the answers one be able to design a diagram for large cardinal tree in implication order and post it as an answer which summarizes the answers of the others.
 A: As an answer to the question (1), one can produce a series of natural orderings on large cardinal axioms using the operator $Con(ZFC+\dots)$.
$\sigma \leq_0 \theta \Longleftrightarrow ZFC\vdash \theta \longrightarrow \sigma$
$\sigma \leq_1 \theta \Longleftrightarrow ZFC\vdash Con(ZFC+\theta) \longrightarrow Con(ZFC+\sigma)$
$\sigma \leq_2 \theta \Longleftrightarrow ZFC\vdash Con(ZFC+Con(ZFC+\theta)) \longrightarrow Con(ZFC+Con(ZFC+\sigma))$
$\cdots$
We have $\leq_0 \subseteq \leq_1 \subseteq \leq_2\subseteq \cdots$ and one can think about an animated diagram which shows the change of positions of each large cardinal axiom in the tree when its ordering varies over $\langle \leq_i:i\in \omega\rangle$.
An important property of these orderings on large cardinal axioms is that by the definitions we have $\forall \sigma,\theta \in \text{Large Cardinal Axioms}~~\forall i,j\in \omega~~((i\leq j~~\wedge~~\sigma=_i \theta )\Longrightarrow \sigma =_j \theta)$ and so maybe there is a (possibly ordinal valued) step $k$ such that all main large cardinal axioms are equivalent in the sense of $\leq_k$ ordering. Note that the same phenomenon happens between $WI$ (existence of a weakly inaccessible cardinal) and $SI$ (existence of a strongly inaccessible cardinal) because we have $WI<_0 SI$ but $\forall i\geq 1~~WI=_i SI$.
Also one can ask some natural questions about the interactions of these orderings with each other as follows.

Definition. Let $\sigma,\theta$ be two large cardinal axioms. Define the convergence number of $\sigma, \theta$ ($\alpha_{\sigma, \theta}$) to be $min\{\gamma\in Ord~|~\sigma =_\gamma \theta\}$ if $\{\gamma\in Ord~|~\sigma =_\gamma \theta\}\neq \emptyset$ and $\infty$ if$\{\gamma\in Ord~|~\sigma =_\gamma \theta\}=\emptyset$.
Example. $\alpha_{WI,SI}=1$
Question 1. Is $\alpha_{\sigma,\theta}<\infty$ for each two large cardinal axioms $\sigma, \theta$?
Question 2. Is there an ordinal $\alpha_0$ such that for each two large cardinal axiom $\sigma,\theta$ we have $\alpha_{\sigma,\theta}\leq \alpha_0$?
Question 3. What is the convergence number of "existence of a strongly compact cardinal" and "existence of a superstrong cardinal"?
Question 4: Consider $\mathbb{L}$ be the set of large cardinal axioms. Is there any ordinal $\alpha$ such that the $\langle \mathbb{L}, \leq_{\alpha}\rangle$ be a linear order?

A: There is also:
c) interpretability strength order:
$\sigma \leq_F \theta \Longleftrightarrow \exists f\ \forall \psi\  ZFC\vdash Con(ZFC+\theta+f(\psi))\longrightarrow Con(ZFC+\sigma+\psi)$
for some suitable interpretation $f$.  This article on Independence and Large Cardinals gives more exposition.
A linear order is also likely here.
If we found $\theta$ and $\phi$ to be incompatible, we would probably say "$\theta$ is a large cardinal axiom, about the size of the set-theoretic universe, and $\phi$ is more about the structure of the universe, so it's not really a large cardinal axiom."
