Bridges - changes for Bridges V2 - relay client part (#4494)

Contains mainly changes/nits/refactors related to the relayer code
(`client-substrate` and `lib-substrate-relay`) migrated from the Bridges
V2 [branch](https://github.com/paritytech/polkadot-sdk/pull/4427).

Relates to:
https://github.com/paritytech/parity-bridges-common/issues/2976
Companion: https://github.com/paritytech/parity-bridges-common/pull/2988


## TODO
- [x] fix comments

## Questions
- [x] Do we need more testing for client V2 stuff? If so, how/what is
the ultimate test? @svyatonik


- [x] check
[comment](https://github.com/paritytech/polkadot-sdk/pull/4494#issuecomment-2117181144)
for more testing

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Co-authored-by: Svyatoslav Nikolsky <[email protected]>
Co-authored-by: Serban Iorga <[email protected]>