You are correct in stating that, if a disposal is made under an unconditional contract, the date of disposal for capital gains purposes will be the date on which the contract is made, or exchanged. It is not the later date when the asset is conveyed.
The information in HS283 is also accurate in that a period of ownership of a dwelling house ends when such a disposal is made.
HS283 Private Residence Relief (2022)
On the assumption that the seller qualified for Private Residence Relief, from the date of their acquisition of the dwelling house until the date they moved out and that they subsequently disposed of the house within 9 months of their leaving.
In such circumstances they would qualify for full Private Residence Relief by virtue of the 'final period exemption'.
CG64985 - Private residence relief: final period exemption