  The construction of initial data for black-hole binaries usually involves the
choice of free parameters that define the spins of the black holes and
essentially the eccentricity of the orbit. Such parameters must be chosen
carefully to yield initial data with the desired physical properties. In this
paper, we examine these choices in detail for the quasiequilibrium method
coupled to apparent-horizon/quasiequilibrium boundary conditions. First, we
compare two independent criteria for choosing the orbital frequency, the
"Komar-mass condition" and the "effective-potential method," and find excellent
agreement. Second, we implement quasi-local measures of the spin of the
individual holes, calibrate these with corotating binaries, and revisit the
construction of non-spinning black hole binaries. Higher-order effects, beyond
those considered in earlier work, turn out to be important. Without those,
supposedly non-spinning black holes have appreciable quasi-local spin;
furthermore, the Komar-mass condition and effective potential method agree only
when these higher-order effects are taken into account. We compute a new
sequence of quasi-circular orbits for non-spinning black-hole binaries, and
determine the innermost stable circular orbit of this sequence.
