For me, Independent means 'independent from authorities'.
This is all about self-sovereignty, it's about becoming a master over one's own life, it's about learning to make decisions and live one's live according to one's truth. This is not to say when one is independent that one doesn't consult other people and get other people's feedback in regards to decisions that one makes. It is about becoming one with oneself and integrating all the skills one has of observation and questioning. It is like being spiritual and a scientist at the same time, one integrates the two parts of reality - what one can observe - the past or the happening and what one can intuit about it by questioning what one has observed and waiting or listening to receive an answer.
So for example, something 'new' and 'unknown' happens in your life, which requires that you make a decision on how to proceed. Because you've never encountered exactly these circumstances before, it is an unknown for you. To be able to proceed and make a good decision, which is in-line with your life and journey, then you make use of what is available to make your decision or conclusion. You observe the happening, you get information about it by doing research, questioning others, consulting experts, etc. Part of this is observing yourself, your reactions & your current life. Then after you've questioned all that you can about the happening, then you will need to make a decision on how to proceed. To do this next step, then you ask yourself, what rings true here, what is it in my heart that I really want to do here, what is the best possible scenario. Because you've never encountered this scenario before, then you can only make 'your best guess', the best decision possible; even it might turn out to be a mistake. But that is how we learn, and the more we rely on our what’s in our heart to make the final decision, then the more we will grow in self-sovereignty and self-reliance, knowing truly that we are in control of our own destiny, that nobody is forcing us to make our decision the way others might dictate.
To contrast this, it might help to know what being dependent means. It means that instead of making your own decision in regards to what to do, you just do what others tell you, or you do what is normal - what the majority is doing. People take this route, because they don't want to accept responsibility in case the decision they make ends in disaster or is a mistake. The benefit of this is that if a mistake happens, you can then give the fault to another person and feel like a victim: you told me to do this, or the doctor misdiagnosed my disease, or the lawyer made a mistake, etc. Then you can take it a step further and sue the people who 'did you wrong'. That's pretty much the society we have today, a bunch of sheep, not taking self-responsibility, going with the crowd, and then when bad things happen they find someone else to blame it on. What a game this is, and all because people are afraid to take ownership of their life and wield their abilities to observe, question and make decisions.