VED exemptions and reductions all stem from the Vehicle Excise and Registration Act 1994 (as amended).
Exemption for disabled people is set out in Schedule 2 (Para 19) and the stipulation for those eligible begins:
"A vehicle is an exempt vehicle when it is being used, or kept for use, by or for the purposes of a disabled person..."
and subsequently continues:
"a vehicle is deemed to be registered under this Act in the name of..." [an eligible disabled person] "...if it is so registered in the name of..." (a) an appointee, or (b) a nominated person.
Reduction for disabled people is set out in Schedule 1 (Para 1ZA) and the stipulation for the 50% rate for those eligible begins:
"This paragraph applies to a vehicle when it is being used, or kept for use, by or for the purposes of a disabled person..."
and subsequently continues:
"a vehicle is to be treated as registered under this Act in the name of..." [an eligible disabled person] "...if registered in the name of..." (a) an appointee, or (b) a nominated person.
https://www.legislation.gov.uk/ukpga/1994/22/contents
It appears to me that use of the car is identical in both cases i.e. not for the personal use of appointee or nominated person.