Ok, this is not really a VCS question, this is first and foremost and architectural problem - how do you structure and build your application(s) in such a way that you can deliver the specific use cases to each hospital as required whilst being able, as you suggest, to fix bugs in commom code.
I think one can state with some certainty that if you follow the model suggested in your image you aren't going to be able to do so consistently and effectively. What you will end up with is a number of different, discrete, applications that have to be maintained separately even though they have at some point come from a common set of code.
Its hard to make more than good generalisations but the way I would think about this would be something along the following lines:
Firstly you need a core application (or set of applications or set of application libraries) these will form the basis of any delivered system and therefore a single maintained set of code (although this core may itself include external libraries).
You then have a number of options for your customised applications (the per hospital instance) you can define the available functionality a number of means:
- At one extreme, by configuration - having one application containing all the code and effectively switching things on and off on a per instance basis.
- At the other extreme by having an application per hospital that substantially comprises the core code with customisation.
However the likelyhood is that whilst the sum of the use cases for each hospital is different individual use cases will be common across a number of instances so you need to aim for a modular system i.e. something that starts with the common core and that can be extended and configured by composition as much as by any other means.
This means that you are probably want to make extensive use of Inversion of Control and Dependency Injection to give you flexibility within a common framework. You want to look at extensibility frameworks (I do .NET so I'd be looking at the Managed Extensibility Framework - MEF) that allow you to go some way towards "assembling" an application at runtime rather than at compile time.
You also need to pay attention to how you're going to deploy - especially how you're going to update - your applications and at this point you're right you're going to need to have both your version control and you build environment right.
Once you know how you're going build your application then you can look at your version control system - @VonC is spot on when he says that the key feature is to be able to include code from shared projects into multiple deliverable projects.
If it were me, now, I'd probably have a core (which will probably itself be multiple projects/solutions) and then one project/solution per hospital but I would be aiming to have as little code as possible in the per hospital projects - ideally just enough framework to define the instance specific configuration and UI customisation
As to which to use... if you're a Microsoft shop then take a good long hard look at TFS, the productivity gains from having a well integrated environment can be considerable.
Otherwise (and in any case), DVCS (Mercurial, Git, Bazaar, etc) seem to me to be gaining an edge on the more traditional systems as they mature. I think SVN is an excellent tool (I use it and it works), and I think that you need a central repository for this kind of development - not least because you need somewhere that triggers your Continuous Integration Server - however you can achieve the same thing with DVCS and the ability to do frequent local, incremental, commits without "breaking the build" and the flexibility that DVCS gives you means that if you have a choice now then that is almost certainly the way to go (but you do need to ensure that you establish good practices in ensuring that code is pushed to your core repositories early)
I think there is still a lot to address purely from the VCS question - but you can't get to that in useful detail 'til you know how you're going to structure your delivered solution.
