If you advocate a model with separate club and team captains, you are asking the holders of the two posts to do two different things. The club captain would presumably be the one that speaks to the media, attends functions on the club's behalf etc etc. Essentially a PR role? This needs to be someone who is media-savvy and also perceived by the media in the "right way" - not so young to be seen as inexperienced, not so old to be seen as past it. Neither too safe and boring nor too fiery and controversial. Must be able to speak with some eloquence and intelligence. In this squad I'd say you are looking at Jagielka or Barry
The team captain carries out a match day role, primarily on the field, and some of the various aspects of that role include handling the referee and leading the team. You also need more than one person capable of taking on this role because your chosen captain may come off injured or be sent off. The main issue for me is that different players have different psychological needs and triggers, so respond in different ways to different type of leadership. Some people need a blood-and-thunder type of captain who will "lead the charge". Others are very capable of self-motivating and may need a calm, cool-thinking captain on hand to ensure they remain in control and focused. The media side of things is far less of an issue, but obviously the team captain has to be in the starting lineup on merit. At the moment the choices are Jagielka, Barry, Coleman and McCarthy. I would personally add Mirallas to the list but it's barely worth mentioning as Martinez is dead set against letting him play and many fans want him gone.