Molecular Hamiltonian of Helium hydride ion
Your provided note is about the HF Hamiltonian in the one-electron basis set (atomic orbitals). Since a minimal STO-3G basis set is used here, the corresponding matrix representation of this Hamiltonian is a $2\times 2$ matrix.

The $3\times 3$ Hamiltonian from the SI is the FCI Hamiltonian in the many-electron basis set. These basis functions are Configuration State Functions, arising from linear combinations of (excited) Slater determinants (electron configurations), which in turn are built from the HF molecular orbitals.

Configuration Interaction calculations (like FCI) are typically done on top of a HF calculation, to correct for missing electron correlation effects. Your linked note is only about HF. For details about CI, I can recommend the book by Szabo and Ostlund which is also cited in your note.