I came across this discussion because I was looking for a solution to the same problem. Eventually I was able to find the answer myself. It's really in the "settings presets" where the default is "Signup is just for invited people, and all users can invite as many people as they like." Trusting, of course, that this is the ideal setting, you would almost never think of selecting "Use custom settings." But upon doing, only then you would see a number of options, and REALIZE that the sole reason why the invite button or the form does not show up to all users aside from the admin - is the "Inviting lockdown" or the number of days a member or user has signed up before he or she can invite anyone. I changed it from the default "30" to "1" and the invite button instantly showed up for all my site users. I think the problem really is miscommunication as the default preset never said anything about the lockdown.