In addition to @ShegitBrahm's aspect of group work, selbständig to me has a connotoation of not only doing things without assistance (eigenständig), but in addition to take the "surrounding decisions" on one's own. E.g.:
- the students do their homework selbständig vs. eigenständig: selbständig has a connotation of not only doing the homework by themselves and unassisted (eigenständig), but it also has a connotation there was no need to remind them to do their homework.
- The same for the shoe-tying child. Selbständig has a connotation of the child having the bright idea on its own to tie their shoes (and then successfully doing so).
But these are only connotations, and you could mostly use both in both occasions.
In my everyday usage, selbständig is far more fequent, eigenständig sounds slightly unusual to me.
For the tribes in 2.3, I'd also use eigenständig rather than selbständig - but I couldn't say why.
If the family 2.2 wants to travel independently, I'd either put "unabhängig" or "einzeln" or "allein" - eigenständig sounds rather elaborate here (?)
For the young man, there may be a bit of confusion with the professional meaning of selbständig:
professional/legal meaning: selbständig
Selbständig has a special meaning when describing the professional legal standing. This is the one meaning of selbständig where you cannot get away with using eigenständig.
The employer can tell their employee (direct them) what, when and where to work (of course within the legal limits like maximum number of working hours per week, etc.).
If you work, but noone can direct you, you are free to decide on your own what you'll do, where you do that and when you do that, then you are selbständig according to labor/social insurance law. If you do your work on your own account and on your own risk and with a long-term aim of financial gains, then your are selbständig according to tax law.
The textbook examples are
- Freiberufler (≈ freelancers) which is a particular group of self-employed whose work is not gewerblich (≈ commercial) such as lawyers, medical doctors, engineers, ... who are not employees.
- Gewerbetreibende Einzelunternehmer (someone running a commercial business) such as salesmen, craftsmen, ... who run their own business.
(All these professions you can also do as employee.)
The CEO = Geschäftsführer in @ShegitBrahm's answer is a bit more difficult: I'll explain with the Geschäftsführer of a GmbH. A GmbH always has (by law) a Geschäftsführer. The Geschäftsführer (CEO) of the GmbH is the one/are the ones who direct/manage the every-day work.
Knowing that someone is Geschäftsführer, however, does not tell you whether they are employed as Geschäftsführer or whether they are selbständig.
The Gesellschafter (owners, shareholders) of the GmbH can hire (employ) someone else as Geschäftsführer (who is then not selbständig*), or they can to that work themselves (selbständig), this is sometimes emphasized by saying they are Gesellschafter-Geschäftsführer or by saying they are selbständig and Geschäftsführer or their own GmbH.
Just like a lawyer may run their own office (selbständig) or be employed by someone else.
*The Geschäftsführer is also employer for the "normal" employees of the GmbH, but that's a different story: the Geschäftsführer (rather than the Gesellschafter) is the one to direct the employees.