Managers like #3 have the ability to advocate for their directs when it comes to career development and advances and when it comes to salary.

The one weakness with good managers who are both technically strong and know how to navigate corporate America and "manage up" seems to be that they don't have the time to be technical leads and get into the nitty gritty of what their department is doing. They have to hire or promote someone as the "benevolent dictator" who enforces best practices and who has interpersonal skills to keep the manager informed and to talk to the customer - either the internal or external customer.

The architect role is really important for a type 3 manager.

If you need a mentor though, manager #1 is better but go in with the understanding that the only way you're going to get ahead in your career with that type of manager is to learn what you can from them and be prepared to advocate for yourself within the organization or more realistically, find another job and use what you've learned.