REST is a methodology for going about implementing an API. Web API is designed to be a RESTful framework but it is up to you to implement it that way. You could go about implementing web api in a non-restful manner and still have it meet your needs.
But you are correct, one of the criteria of being RESTful is to implement HTTP verbs which is what web api allows you to do very easily.
I would suggest you read up more on what REST is and not over complicate the issue. Web API is fully capable of creating a RESTful API but once you grasp the concepts of REST itself, then you'll be able to create the API in that manner.
What is REST?[
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Representational state transfer - Wikipedia[
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http - What exactly is RESTful programming? - Stack Overflow[
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Do you know what a REST API is? — SitePoint[
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