I could go on an on about the differences between an Engineer, a Tech, a Manager, and a Team lead. It sounds like what you are looking for in a manager is really a team lead position.
Formally, you could be right. Informally, both the team leader and manager hat usually end up on the same person, even if he lacks one title or the other. If you haven't got a team lead it's pretty obvious, if you do have a team lead then in my experience the manager does the HR/administrative bits and leave the actual work management to the team lead or the project manager if you work on a project.
For example, with no formal title I basically had the responsibility to:
1) Execute the actual project
2) Delegate as possible to the two juniors
3) Support the two juniors
4) Train the two juniors
Sure, there was a project manager dealing with the contract and formal contact with the client. There was a manager dealing with formal HR bits. But I felt I was a bit project manager, a bit team lead, a bit manager and a bit mentor all at once. It was a constant prioritization between:
1) What must I do to get the project done?
2) What can I delegate to free up my time?
3) What should I delegate to teach them?
4) What should we walk through together?
When you're in practice managing 100% of their time, you get all the hats whether you want to or not.