Independent Councillor - Such A Thing?

I ask this question simply because I cannot find one on the East Lindsey District Council and this has always been something that I am confused by.  

My view of an Independent Councillor is one who is free from party allegiance and is free thinking, allowing them to vote independently of anyone else. I think that is a logical view. This definition would back up that view:

Politics. a person who votes for candidates, measures, etc., in accordance with his or her own judgement and without regard to the endorsement of, or positions taken by, any party.

Now as I say I cannot find such a being in the ranks of ELDC councillors. Nobody actually listed as Independent. We have East Lindsey Independent Group and we have Independent/Liberal Democrat, but no actual Independents. 

Firstly I have no idea what an Independent/Liberal Democrat actually is but it certainly does not read as Independent. Secondly. If you are Independent then why would you join a group? Surely once you have joined a group then you are part of that group and are therefore, by definition, not Independent? The leader of the group, yes there is one, strange isn't it, is responsible for representing the group and responsible for allocating councillors to particular roles and committees within the council. Just as the leaders of the party groups do. So what is the difference between a party group and an Independent group? Nothing in my eyes. Are the group members going to vote on items independently?

To cap this off I would refer to an email I saw sometime ago when an Independent councillor informed the council that he was no longer and Independent but would be joining the Independent Group. Confused? 
As I said at the beginning. So am I.