Understanding UK split years and double taxation agreements

From Residence, Domicile and Remittance Basis Manual 12010:

If during a year the individual starts to live or work abroad, or comes from abroad to live or work in the UK, the tax year will be split into 2 parts, if their circumstances meet specific criteria:

• a UK part for which they will be charged to UK tax as a UK resident
• an overseas part for which, for most purposes, the individual will be charged to UK tax as a non-UK resident.

Then, from Residence, Domicile and Remittance Basis Manual 12020:

An individual must be UK resident for a tax year under the SRT to meet the criteria for split year treatment for that year. They will not meet the split year criteria for a tax year for which they are non-UK resident under the SRT.

SRT stands for Statutory Residency Test and, as I previously expounded in this thread, I pass the test and I am indeed a tax resident for 2019-2020, even if only lived there from 6 April 2019 to 1 August 2019.

This is the part from Document 12020 that confuses me:

Split year treatment will not affect whether the individual is regarded as UK resident for the purposes of any double taxation agreement.

Isn’t the whole point of the split-year treatment to affect the status of UK residency during the overseas part? That is, one is indeed a non-UK resident for that period of time?