You must have seen SMS's of different types without realizing it. e.g. alert messages that auto-popup on the screen, commissioning messages that you get from your telco (they don't show up on the screen; they set your phone parameters for you), among other types.
The WAP Push message doesn't have anything to do with paid services. The relationship is only casual as it is primarily used to promote paid services. My own systems send out WAP push messages for a number of reasons.
I haven't come across phones that can send WAP Push messages, although you can write a program to do so. I have and it works.
Basically, if you send a URL inside a regular text message most older phones will simply display the text as is without making it a hyperlink. The newer operating systems of phones would convert it into a hyperlink.
However, whoever designs standards, designed a specific format for using when links are to be sent to phones. Every phone that has Internet capability is supposed to understand it. Except for Android phones, for some unknown reason. This type of message contains a URL with a title only. When you open the message and press the appropriate phone button it launches the browser on your phone and takes you to that page.
