Public = Any user can join and any user can see it
Approval = Any user can join and any user can see it, but their join request must be approved
Invite = Users can only join if they're invited and they can only see it if they're invited

In addition to the above types there is also Access. If the user doesn't meet the access they won't be able to see or join the group at all.

If you only want Public group types then within GJ > Config > Groups > Defaults and set "Type" to "Open" then next to it set it to "Hide" so all groups created will always be public. Note as a CB moderator you won't see any change; test with a default registered user to see what they'd see.

If you want to limit what types of groups a user can create then this is done on a Category by Category basis. Simply edit your category and select Open and Approval from the "Types" parameter as needed. You can also set a default for this within GJ > Config > Categories > Defaults.

Could you please also confirm that is I allow "approval" groups, if the creator of the group defines "not public" for the wall then the wall posts won't show in groupjive except the user is member of the group and also that the activity plugin won't show wall posts ?

Correct, if a group is Public or Approval where users can freely view the groups and you set a tab to be not public then that tab and its activity will not be visible to users that have not joined the group. Note even if they can see the Wall tab they can not post on it unless they're members of the group if that's your concern.