This paper presents a search for direct top squark pair production in events with missing transverse momentum plus either a pair of jets consistent with Standard Model Higgs boson decay into $b$-quarks or a same-flavour opposite-sign dilepton pair with an invariant mass consistent with a $Z$ boson. The analysis is performed using the proton-proton collision data at $s=13$ TeV collected with the ATLAS detector during the LHC Run-2, corresponding to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. No excess is observed in the data above the Standard Model predictions. The results are interpreted in simplified models featuring direct production of pairs of either the lighter top squark ($t~1$) or the heavier top squark ($t~2$), excluding at 95% confidence level $t~1$ and $t~2$ masses up to about 1220 and 875 GeV, respectively.