I have replaced my old Mac Mini with new one. What should I do about my Time Capsule?

Question

With the old Mac Mini (2009 model) I used Time Machine and a Time Capsule to back up everything on the Mac Mini itself and on external hard drive I added when I needed more storage space.  I've replaced the old Mac Mini with a 2011 model, and migrated the contents of the old one to the new one from Time Machine. 
I'm now trying to figure out what I should do to my Time Capsule and/or Time Machine to make backups going forward of the same content, now on my new Mac Mini and same external hard drive. 
When I go into Time Machine, it does not seem to recognise the items on the external hard drive as ones that I can backup (or not, as there were some items there that I did not and do not wish to backup to Time Capsule) through the new Mac Mini. 
I am uncertain too whether the Time Machine/Time Capsule will recognise the items of my old Mac Mini as the same ones that are now on my new Mac Mini and save changes there, or whether it will try to save all over again all the same items, as they are now on a different computer. 
My inclination is to clear out the contents of the Time Capsule now and start all over again.
I have looked through information provided by Apple but not found answers to my questions.

Answer 1

The situation concerning the hard drive is complicated.  It is not included in the list of Excluded Items.
If the drive is not "Excluded", then it must be included.  That is the way that Apple's software works.
But, nor am I given the option to select it or any of the items in it for adding to the list of Excluded Items.
If you want to add an item to the list of Excluded items, click the + button at the bottom of the Excluded list, then navigate to select the drive to add it. Once the drive has been added, you can go back in and select the items.
When I go into the exclude from backup area and select the hard drive, it and the items in it appear in dimmed form, but I cannot select them.
Sorry, I do not understand what you are trying to say here.
What I want to do, but cannot, is what I did before, select one of the items in it not to be backed up and leave the rest to be backed up
I have no idea how you were able to do this. As long as I can remember, Time Machine will not allow you to pick and choose items on a drive to be backed up or excluded. Your option is to have the entire drive backed up, or the entire drive to not be backed up.
have not seen the 'Quick Erase' option.
You would use AirPort Utility to erase the Time Capsule drive. Open AirPort Utillty and click the Disks tab at the top of the screen. Then, click Erase Disk.  The Quick Erase option will appear.