I like the term "Serious and Competitive club' as it suggests that people are there to play seriously... that they are going to take their time, declare/defend properly, tank if necessary, etc. Maybe be descriptive and explain that you're expected to play properly, take your time, etc. That people go there for good practice. Not meant to exclude people who aren't experts, but people who won't take things seriously. (Skill classifications can be done within the section... set table for "Beginner or Intermediate only", or "Adv+", or whatnot).
And I really don't think there should be further classifications. I think the "Main Bridge Club" should stay for everything else. It can be renamed "General Bridge" or "Relaxed/friendly" or whatever. (My favourite is "Relaxed bridge club")
Further distinguishments such as "Fast Bridge", "Precision only", etc, should probably be made with the table description field instead of a different club.
The problem is if you try and fragment the bridge populace into categories, such as Beginner's club - Intermediate club - advanced club - expert's club - friendly club - fast club - serious club, etc, then you get a mess. No one knows where the line is drawn and people will move to the extremes. There should be a main club for everything, and different clubs to cater for 'outliers', rather than to try and 'categorise' the bridge playing populace.
Or we can leave it the way it is, because it's fine!
The difference between theory and practice is that in theory, there is no difference between theory and practice, but in practice, there is.