Intuition about what makes an algorithm a SVM

I was going through the stanford readings. Specifically SVM

I understood all of it including the Maths but at the end there was this “Aside: Other Multiclass SVM formulations” that confused me a bit. It mentions that there are different ways to do SVM. I thought I have learned one algorithm. Now if I look at another algorithm how will I know whether this is SVM.

Then I thought about this intuition presented in the beginning

The SVM loss is set up so that the SVM “wants” the correct class for each image to a have a score higher than the incorrect classes by some fixed margin Δ

Maybe I can just replace the SVM with algorithm and take that as a take away intuition of what makes any algorithm a SVM?