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](https://arxiv.org/abs/1307.3486)* or *[otherwordly](https://jdh.hamkins.org/otherwordly-cardinals/)*; 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](https://arxiv.org/abs/2107.01580)*, *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](https://doi.org/10.1016/j.apal.2007.02.003)*. 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](https://doi.org/10.1016/j.apal.2007.02.003); $P-E_0$ 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.*