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I am amazed at seeing that the limit is defined by continuity. It is interesting.

99. Expository notes

Common practice in calculus books is to define continuity using limits. I define limits using continuity; continuity is a simpler concept.

Are they equivalent? Or both systems work smoothly?

Continuity in the topological sense implies continuity by limits. For topological spaces with a countable (local) basis, the converse is also true.

So, in general it's not equivalent. For the reals etc., it is.

They are equivalent if you consider limits of nets rather than limits of sequences.

They are logically equivalent for the reals. Search the web for [limits "in terms of continuity"].