Now will these 10 experiments be of different names? How will I know these are part of the 'mnist1' HPO case?

Yes (they will have the specific HP name/value combination).
FYI names are not unique so in theory you could have multiple experiments with the same name.

If you look under the Configuration Tab, you will find all the configuration arguments for the experiment. You can also add specific arguments to the experiment table (click the cogwheel at the right top corner, and select +hyper-parameters)