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I know several ways to modify large cardinal notions:

If a family of large cardinal notions is defined with an ordinal parameter (call it $\gamma-A$), one can define the property of being $\gamma-A$ for all $\gamma$:

• If $\kappa$ is $\gamma$-shrewd for all $\gamma$, it is said to be shrewd.
• If $\kappa$ is $\gamma$-strongly unfoldable for all $\gamma$, it is said to be strongly unfoldable (equivalent to being shrewd).
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong.
• If $\kappa$ is $\lambda$-supercompact for all $\gamma$, it is said to be supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible.

If the definition of a large cardinal notion $A(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a large cardinal notion asserting that there are unboundedly many such $\theta$:

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be 0-extendible or otherwordly; if additionally $\kappa$ is inaccessible, it is said to be 0-pseudo-uplifting; and if additionally $\kappa$ and $\theta$ are both inaccessible, it is said to be 0-uplifting. If there are unboundely many such $\theta$, $\kappa$ is said to be totally otherwordly, pseudo-uplifting or uplifting, respectively.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. If for every $\gamma$ and every $A \subseteq V_\kappa$ there exists such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be superstrongly unfoldable.
• The definitions of superstrong, almost huge and huge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and certain other properties. If for every $\gamma$ there is such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be globally superstrong, super-almost-huge, or superhuge, respectively. One can define a similar strengthening of the definition of $\gamma$-extendible cardinals (there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$); this strengthening doesn't appear to have a name but one could call such cardinals globally $\gamma$-extendible.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. If for every $\gamma$ there is a high jump embedding $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be super-high-jump. One can similarly define super-high-jump for strongness cardinals but this property is equivalent to being globally superstrong just like high jump for strongness is equivalent to superstrong.
• An elementary embedding $j: V_\lambda \to V_\lambda$ is called a rank into rank, $I_3$ or $E_0$ embedding. A rank into rank embedding satisfying certain additional conditions is called an $E_n$ embedding (for $n \lt \omega$. An elementary embedding $j: V \to M$ such that $j^n(\kappa) \subset M$ for all $n \lt \omega$ is called an $I_2$ embedding (any $I_2$ embedding restricts to an $E_1$ embedding and conversely any $E_1$ embedding extends to an $I_2$ embedding) and the critical point of an $I_2$ embedding is sometimes said to be $\omega$-fold superstrong. A further strengthening of $E_n$ is called $I_1$ or $E_\omega$. If for every $\gamma$ there is an $E_n$ embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be a $P-E_n$ cardinal; $P-E_0$ cardinals are also called $\omega$-fold extendible. If for every $\gamma$ there is an $I_2$ ($\omega$-fold superstrong) embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be $\omega$-fold strong (which is of course equivalent to $P-E_1$). Similiarly we can define $P-E_\omega$ cardinals.

If the definition of a large cardinal notion $A(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a $C^{(n)}$ variant, additionally asserting that $\theta$ is $\Sigma_n$-correct.

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be otherwordly. One can define $C^{(n)}$-otherwordly cardinals by additionally requiring that $\theta$ is $\Sigma_n$-correct.
• The definitions of superstrong, almost huge and huge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and the definition of $\gamma$-extendible cardinals says that there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$. If additionally $\theta$ is $\Sigma_n$-correct, $\kappa$ is said to be $C^{(n)}$-superstrong, $C^{(n)}$-almost huge, $C^{(n)}$-huge, of $C^{(n)}$-$\gamma$-extendible, respectively.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $C^{(n)}$-high jump for strongness and $C^{(n)}$-high jump cardinals by additionally requiring that $\theta$ is $\Sigma_n$-correct. Just like high jump for strongness is equivalent to superstrong, $C^{(n)}$-high jump for strongness is equivalent to $C^{(n)}$-superstrong.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. One can define $C^{(n)}$-weakly superstrong cardinals by requiring that for every $A$ there is such a $j$ with $j(\kappa)$ $\Sigma_n$-correct.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If $M^{j(f)(\kappa)} \subset M$, $\kappa$ is said to be Shelah for supercompactness. One can define $C^{(n)}$-Shelah cardinals as follows: for every function $f: \kappa \to \kappa$ such that all ordinals in the range of $f$ are $\Sigma_n$-correct in $V_\kappa$, there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$ and $j(f)(\kappa)$ is $\Sigma_n$-correct. One can similarly define $C^{(n)}$-Shelah for supercompactness cardinals.
• This method can be combined with those described above to define $C^{(n)}$-totally otherwordly, $C^{(n)}$-pseudo-uplifting, $C^{(n)}$-uplifting, $C^{(n)}$-superstrongly unfoldable, $C^{(n)}$-globally superstrong, $C^{(n)}$-extendible, $C^{(n)}$-super-high-jump, $C^{(n)}$-super-almost-huge, and $C^{(n)}$-superhuge cardinals.

To be continued.