Marc, the UK Gov website cannot vary law, and it is frequently very wrong in its interpretation of law. If there is a contradiction between law and the UK Gov's interpretation of the law, how on earth would UK Gov guidance take precedence?
That said, in this case the UK Gov guidance seems correct if vague. Its your interpretation of the UK Gov that's wrong. The UK Gov guidance you have quoted refers specifically taking a vehicle away from an MOT test to fix defects which have caused an MOT fail, which is consistent with the law. Not for driving a vehicle to a garage for repairs which have not caused an MOT fail, as is the case for the OP.