Not clear to me how the Manifest.toml are used. In python project, I am used to commit the .lock files so that it is possible to define a default configuration for the tests or to ensure to get reproducible results.

A Manifest.toml file is created when you first install the project, for example by doing:

julia --project
using Pkg
Pkg.instantiate()

If a manifest file exists, Pkg.instantiate() installs exactly all dependencies listed in the manifest.
In contrast, Pkg.update() looks at the Project.toml file and updates all dependencies to the newest versions allowed by the Project.toml file.

In the last commit I added two default manifests that document the last tested set of packages used, Manifest-v1.10.toml.default and Manifest-v1.11.toml.default for Julia 1.10 and Julia 1.11.

Normally these are not needed, Project.toml should be sufficient, but if something brakes (because a minor package update was buggy) you can always do

cp Manifest-v1.11.toml.default Manifest-v1.11.toml
julia --project
using Pkg; Pkg.instantiate()

And you have the last know good set of packages installed.

Is it clear to you now?

Yes clear, but I feel like only one version could be supported as default. The julia version number is integrated in the Manifest.toml, so this should be sufficient, avoiding to have to rename the file ? After, you may save other manifest in subfolder for example in you have published a study (but I think this should not be in a package, but wise if the publication topic is the package).

No, renaming of the file is needed. If you start Julia, it looks first for a Manifest file that has its own version number in the file name. If that is found, it uses it. If there is no such file, it uses Manifest.toml (without version number). And if there is no such file it creates it.

We always want to support two versions, the latest Julia version with long term support (which is 1.10), at the latest stable version (after we found it mature enough). Before making a release I always run both the unit tests and the examples on both Julia version. Switching the Julia version is easy, for example by executing:
juliaup default +1.10
Ok, once you need to do juliaup add 1.10.

If you only have one Manifest.toml file without the version number in the file name and switch the Julia version you are using, then things get messed up.

Yes clear, but I feel like only one version could be supported as default. The julia version number is integrated in the Manifest.toml, so this should be sufficient, avoiding to have to rename the file ? After, you may save other manifest in subfolder for example in you have published a study (but I think this should not be in a package, but wise if the publication topic is the package).