I think you misinterpreted the parent post. You two are almost certainly talking about the same kind of REST aka the mainstream understanding of it popularized by APIs like Twitter, Twilio, Stripe, etc. The "real/pure REST" folks are talking about by-the-dissertion Fielding's REST which is far more complicated and confusing and is probably not a good idea for a public API. It goes beyond verbs and pretty URLs. Look into Hypermedia APIs. That's the One True REST.
Yeah, Twilio's API isn't REST as it has versions and content types in the URIs. Plus, you're supposed to construct URIs (and all the clients do this AFAIK, since it'd be idiotic waste of a roundtrip not to). E.g. "/2010-04-01/Accounts/{AccountSid}/IncomingPhoneNumbers/{IncomingPhoneNumberSid}"
Yet it's the closest thing I've seen held up as a "good RESTful API". In practise, it just means I've got to write more code to shove parameters into the URI and body.
"REST" as used by pretty much everyone is just a codeword for "we didn't use SOAP!". And maybe the URLs are more than just "/FooService" and maybe they use HTTP verbs.
