Is this correct that if I decide to setup an own TravisCI agent, I will be enabled to be independent of the existing distributed Travis computation network?
Travis CI doesn't offer running your own workers, you use their infrastructure unless you mean the on-prem Travis Enterprise. You might be thinking of GitLab CI?
Travis CI is build as a cloud solution focusing upon github repositories. It is possible to setup a Travis CI in-house as a self-hosted solution, however the open-source project is not build with this in mind and therefore it is likely that you will hit limitations. This is why Travis CI has an Enterprise solution: https://enterprise.travis-ci.com/. I think you would be better off looking at similar CI products or buying the enterprise edition of Travis which are focused upon you owning both the master server and the build agents. However, it depends on your specific need.