These are not equivalents. Input type=search can have side effects in that it renders the input box differently, for example with a "X" inside to clear the search. Which may be good, but it may also conflict with your design system.
Yes; personally I’d rather have a type=search to have the semantics and then apply CSS on it, rather than a type=text with no semantic meaning at all that’s modified to behave like a type=search.
