It would be easier for you if you just read the summary on the link, and yes, I did actually do that.
To keep it simple, President/VP are synonymous from a document definition and process perspective. All docs are owned by the US government, per law. Any personal docs (which is clearly defined) need to be separated prior to leaving office, NARA will help with this. If the Pres/VP want access for memoirs or for a library after leaving office, NARA will provide the docs in a location which NARA is responsible for securing and which NARA retains control over docs and access.
WRT Obama, for example
Did President Obama take Presidential records to Chicago after he left office?
No. When President Obama left office in 2017, NARA took physical and legal custody of the records of his administration in accordance with the Presidential Records Act. NARA made arrangements to move the roughly 30 million pages of paper Presidential records of the Obama administration to a federally acquired, modified, and secured temporary facility that NARA leased in Hoffman Estates, IL, which meets NARA’s requirements for records storage and security. NARA moved the records to Hoffman Estates because of the intention of President Obama to build a Presidential Library in the Chicago area.
Subsequently, former President Obama decided not to fund, build, endow, and donate a physical Presidential Library to NARA (his foundation is building a privately operated Presidential Center that will not have archival storage for any Presidential records). Instead, the Obama Foundation agreed to help pay for the cost of digitizing the unclassified paper records and for the cost of moving the classified and unclassified records from NARA’s temporary facility in Hoffman Estates to other NARA-controlled facilities (for which NARA otherwise would have to pay). A September 2018 Letter of Intent from the Obama Foundation to the Archivist of the United States addresses Obama’s commitment to paying for these costs; but it in no way suggests that Obama had physical custody of any Presidential records. As NARA stated in September 2022, neither former President Obama nor his foundation “had possession or control over the [Presidential] records” of his administration.