There is not much difference between all the forms of transport that you list: all of them require a transporter room and its equipment, most notably a transporter pattern buffer, a device that stores all information about the dematerialized matter before streaming it to its destination.
In example, transporting from a transporter room to an external site required dematerializing the matter present on the transporter platforms, storing the matter stream into the pattern buffer, then transporting it to the destination site; transporting to a transporter room, was effectively the opposite process.
Site-to-site transport, was not that different: it dematerialized the subject from its location (outside the transporter platform), streamed the matter information to the pattern buffer, then streamed again all this information to the destination site.
According to the TNG Technical Manual:
Site-to-site transport. This refers to a double-beaming procedure in which a subject is dematerialized at a remote site and routed to a transporter chamber. Instead of being materialized in the normal beam-up process, however, the matter stream is then shunted to a second pattern buffer and then to a second emitter array, which directs the subject
to the final destination. Such direct transport consumes nearly twice the energy of normal transport and is not generally employed except during emergency situations. Site-to-site transport is not employed during emergency situations that require the transport of large numbers of individuals because this procedure effectively halves the total system
capacity due to minimum duty cycle requirements.
So, basically, a site-to-site transport is a sequence of a beam-up immediately followed by a beam-down, where the object is not re-materialized and re-dematerialized in between. It is not always used because it requires a greater amount of resources.
You can thank about it like your laptop (= a Transporter room) with two USB pendrives attached (= site A and site B): if you want to move files between one of the to the other, you don't need to physically copy them to your hard-drive (= materialize and dematerialize again), but you can directly copy them (= site-to-site transport) even if you still need the processing power and the RAM of your laptop (= trasporter equipment and pattern buffer).