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Use rust-native errors in `DAGCircuit` (#16134)

* Use rust-native errors in `DAGCircuit`

This should be step 1 for splitting `PyDAGCircuit` from `DAGCircuit`.
The following commits assure that any rust accessible method for `DAGCircuit` that:
1. Does not take a `py` token;
2. Is a public method.
does not need to depend on `PyErr` to process any errors.
It is ensured that these errors:
- Maintain the level of detail that the previous ones did.
- Correctly map to the same Python exceptions as before.

The enumeration for these errors is called `DAGCircuitInnerError` (subject to change) and is implemented with `thiserror`. I chose this over `anyhow` because it is better for specifying where the error comes from based on its variant rather than the message it contains.

* Lint: Use names instead of underscores

* Add: `anyhow!` usage for `DAGCircuitError
- Use `anyhow!` for any error types that don't require tracking of any variables coming from the `DAG` or that are part of the function arguments.
- Consolidate the amount of `DAG` error variants down from 40+ to 21. Can be consolidated further.

* Fix: Existing error formatting
- Use `this_error` to fix the formatting of current errors causing conflicts with their messages.
- Add `is_var` argument to differentiate Variables and Stretches in `VariableNotInlined`.

* Fix: consolidate `RegistryAbsentObject` error.

* Add: errors for `size` and `depth`.

* Refactor: `DAGCircuitInnerError` to `DAGError`

* Fix: Add custom `From` impl for `Infallible`
- The new implementation elides `Infallible` by using a 0-variant `match` which will be valid only with `Infallible` and return nothing.

* Fix: Add context to `WireOutOfRange`
- Remove usage of `anyhow` for `WireOutOfRange`.

* Remove: `anyhow` in `General errors
We removed `anyhow` from the circuit crate and instead assumed any error of type `String` should be assigned to a `General` error.

* Rename: `RegisterRef`, `ConcreteWire` to `RegisterType`, `Shareabl... (continued)

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