Hard vs Soft Margin SVM interpretation

Hello, I have a doubt about the interpretation I should give to hard/soft margin SVM formulations.
If I have correctly understood, in the hard margin we know that the data are linearly separable and therefore we look for the hyperplane that allows us to have the larger margin possible.
Concerning the soft margin interpretation, on the contrary, we said that the margin represents the region where we are not certain of how the data point considered should be classified, so, in a certain way, we want to minimize the margin, so that we are as certain as possible in defining the label of our data points. Is that right? Because I'm not sure this interpretation is correct.
Thank you in advance for your kind answer!

Wasn’t the soft margin related to the presence of a regularisation parameter?

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