I got the answer to my question. For future reference, this is it:

The messages sent by banks, service providers etc are sent via SMS

whereas

The messages by other peers are sent via RCS

RCS is an internet-based messaging protocol (like WhatsApp) that works through Google's Messages app and similar platforms. Unlike SMS, RCS messages are not processed through the standard SMS gateway, which means HTTPSMS (and others) won't detect them. When the sender's internet is off, their phone falls back to SMS, allowing HTTPSMS to capture it.

So, the solution is to ask the sender to switch off their internet before sending messages