Let's clear up any misunderstanding about wiring - what you see feeding your NTE5a is exactly what you might expect to see where the drop-wire or underground feed terminates elsewhere in the building, and standard 3 pair CW1308 has been used from that NTP to your NTE5a. Your telephone line only consists of 1 connected pair.
Worth asking though, this is the only telephone socket connected in the property is it?
The "issue with the bell-wire" normally arises with what ever the user has connected OR plugs in, on their side of the NTE5a.
Regular phone type extension leads, such as what it sounds as if you are using contain a bell wire!
As for in your room, there is little point in fitting a filtered faceplate if you are then going to have trailing extension leads all over the place. In any event when testing from a test socket, a filter should plug directly into the test socket and the modem/router direct into the filter with no extension leads otherwise you only recreate exactly what you had before. The only point in removing the faceplate is to isolate any internal hard wired extensions connected to it.
The only point of fitting a filtered faceplate is if you are going to have the modem/router or the phone right next to it, the latter to save a dangly filter and you then have proper hard wired unfiltered extension wiring for the modem/router.
The most obvious and neatest solution is to fit your own hard wired extension, you can use a filtered extension socket if you don't want a dangly filter at the extension, but performance wise, there is very little or no difference and added expense with a filtered socket. Fitting a standard hard wired extension using the correct cable will not give performance issues and is also the cheapest solution. You can then use the original very short modem/router cable (or if you really need a slightly longer one, get a twisted pair screened one),
For the extension you will need to buy some 2 pair
CW1308 cable, some 3.5mm cable clips and a Disposable IDC tool from the tools page.
I wouldn't recommend buying the cable anywhere except from a reliable source, there is fake standard CW1308 about. You can get the standard extension socket itself at any DIY shed, local store etc. or even eBay, Amazon etc. A standard 86x86mm socket and mounting box should cost no more than £2-3, any more is a rip-off (watch for postage costs if buying online, some are a rip-off as well!).
When connecting the extension you only use the one pair, Blue+white trace on terminal 2 and White+blue trace on terminal 5.
