My latex table is still not centered, even after using centering or begin{center}

If tablenotes is from threeparttable, you should probably use threeparttable. The optional argument for caption is for the LOT. Tablenotes requires items.

documentclass{article}
usepackage{caption}
usepackage{booktabs}
usepackage{threeparttable}
usepackage{pifont}
usepackage{showframe}% MWE only

begin{document}
begin{table}[htbp]% [!] almost never has any effect.
centering
begin{threeparttable}
captionsetup{justification=centering}% which has no effect on single ine captions.
caption[Cake cutting]{Summary of the cake-cutting procedures.}
label{tab:procedures}
begin{tabular}{lccccc}
    toprule
    Procedure                      & Agents & EXP1 & EXP2 & EF & Prop 
    midrule
    Cut and choose (2ACC)                & 2      & ding{51}    & ding{51}    & ding{51}         & ding{51}            
    Cut middle          (2SCC)           & 2      & ding{51}    & ding{51}    & ding{51}         & ding{51}            
    Selfridge--Conway  (3SC)  & 3      & ding{51}    & ding{51}    & ding{51}         & ding{51}            
    Dubins--Spanier ($n$DS) & $n$      & ding{51}(3,4)  & ding{51}(2,3)  & $times$        & ding{51}            
    Knaster--Banach ($n$KB) &$n$      & ding{51}(3,4)    & ding{51}(3)    &     $times$   & ding{51}            
    Even--Paz   ($n$EP)                    & $n$      & ding{51} (4)&$times$  & $times$        & ding{51}           
    bottomrule
end{tabular}
begin{tablenotes}
  item footnotesize EF refers to envy-freeness. Prop refers to proportionality.
end{tablenotes}
end{threeparttable}
end{table}
end{document}