In the bottom left, why does the BM diagram start at -4kNm rather than 4kNm? I understand that if you make a cut you yield this result but if you don't make a cut how would you know whether it starts at -4kNm or 4kNm?
1) I think you might be confusing the moment applied externally to a body with that of an internal bending moment. An internal bending moment is usually given a sign convention that depends on whether it causes tension on the top of a beam or on the bottom of a beam. In this specific case the convention adopted here is a bending moment is +ve if it causes tension on the bottom of the beam.
2) You can apply any convention as long as you are always consistent throughout. So if you define a clockwise moment as +ve than that is perfectly fine. Just don't switch back to a -ve clockwise convention later. That being said, you are right that a standard convention used by many is to assume a clockwise moment is -ve.
If you have device work on Android, you can download Shear Forces Bending Moments app from Google play. This is a link: https://play.google.com/store/apps/details?id=com.hakan.mnq
This app finds shear forces and moments functions by using the method of sections and draws diagrams of these functions for isostatic straight beam. Calculation steps are shown in orderly.
Nugso, the sign of a bending moment is not related to whether it is clockwise or anticlockwise. After all, if you were to go round to the other side of the beam, it would still be bending the same way. The test is whether the effect is a hogging or a sagging effect. Sign conventions vary around the world rather more than you would think, and there are fierce defenders of some of them. Think of whether the tension is on top (hogging) or underneath (sagging). If you are dealing with a column, then you will have to choose point of view (just as you do for a beam, generally viewed from underneath.