I don't want them to go through my data for advertising purposes or to sell my personal data to advertisers (often called "partners" in privacy policies). I also don't want them to sell my personal information to foreign governments (like Facebook selling personal information on US tech users to Chinese tech companies).
I am fine with a privacy policy that has a section for using my data to provide the service (Service Providers section) and to share my data with companies that they use to provide services (as long as those companies must adhere to the same restrictions). As per your example of Twilio, that is fine as long as their contract with Twilio ("trusted business partners are subject to confidentiality agreements") states that Twilio won't sell my phone number to advertisers.