A very warm welcome to the forum!
Take a look at
Kitz’ diagram of the local network. If someone is getting ADSL2+ or the original G.DMT (‘ADSL1’) then the modem at the remote end that they talk to is in the exchange not in a green cab. The speed is up to 20 Mbps downstream.
If you are lucky enough to have the much faster VDSL2 that is FTTC meaning that the modems that you talk to are in a green cab, much nearer to your property, which makes it up to 80Mbps depending on distance and on crosstalk which is the interference caused by leakage of other users’ signals into your copper. In FTTC it is a copper line from the green cab to your house, not fibre as the lying adverts might imply.
Now if you change from ADSL to VDSL2/FTTC, then the change is made in the exchange, because that is where the ADSL modems are. The copper link from the exchange to a roadside box called the PCP is replaced by a fibre link to a different green box called errm, an FTTC cab, don’t know. The FTTC cab is linked by a cable to the PCP by copper and then there is more copper in the PCP which goes to your house. So in the PCP wiring is changed to link the line from your house into the cross connection link to the FTTC cab’s modems. So two changes.
Some unlucky people such as me myself have no PCP cabinet but just direct copper from the house to the exchange - a so-called EO line (‘exchange only’), and have no chance of getting FTTC, not unless Openreach retires the network. So for us it is not like the diagram.
Hope I got that right and hope that helps.
I don’t know what that mysterious grey box is, but someone here will know.