If I send a URL directly it will be hit from my phone. Big difference there.
Messages on my iPhone shows me link previews for links sent to me via SMS, not just via iMessage. I just checked to make sure. Those are necessarily generated on the recipient's end.
This could be due to it being sent from someone in your contact list: https://support.twilio.com/hc/en-us/articles/360013199334-Ho...