For there to be a color singlet gluon color theory would have to be a U(3) theory, not SU(3), but the great weight of experimental evidence assembled over many years supports SU(3) not U(3). In the 60's (before the Standard Model had reached maturity) a couple papers by well-known theorists appeared investigating the possibility of U(3) rather than SU(3) symmetry, but that's about it. One of the papers worked out a scheme for integer rather than fractional charge for quarks, but none of the main ideas ended up in the Std Model, which is still firmly SU(3) => no singlet gluons. In addition there is no experimental evidence for any long range component to the strong force - quite the opposite. There are plenty of unaswered questions in physics and I suspect that most theorists feel that (if they even give the ninth gluon a moment's consideration at all :-) ) there are many other topics to research that are more likely to lead to interesting new results.
Texts on QCD don't divide the generators of $SU(3)$ – and therefore "bicolors of gluons" – into two groups because this separation is completely unphysical and mathematically artificial (basis-dependent).
Moreover, the number of "bicolors of gluons" i.e. generators of $SU(3)$, the gauge group of QCD, isn't nine as you seem to think but only eight. The group $U(3)$ has nine generators but $SU(3)$ is the subgroup of matrices with the unit determinant so one generator is removed. At the level of gluons or Lie algebra generators, the special condition $S$ means that the trace is zero. So the combinations
$$ A(r\bar r) + B (g\bar g) + C(b \bar b)$$
are only allowed "bicolors of gluons" if $A+B+C=0$. Now, in this 8-dimensional space of "bicolors of gluons", there are no directions ("bicolors") that are better than others. For any direction in this space, there exists an $SU(3)$ transformation that transforms this direction into a direction non-orthogonal to any chosen direction you choose. This is true because the 8-dimensional representation is an irreducible one (the adjective "irreducible" means that one shouldn't try to split it to two or several separated collections!). And there doesn't exist any consistent Yang-Mills theory that would only contain the six off-diagonal "bicolors" because the corresponding six generators aren't closed under the commutator.
The actual calculations of the processes with virtual gluons ("forces" between quarks etc.) therefore never divide terms to your two types because this separation is just an artifact of your not having learned group theory. Instead, all the expressions are summing over three colors of quarks, $i=1,2,3$ indices of some kind, and there is never any condition $i\neq j$ in the sums because such a condition would break the $SU(3)$ symmetry.
Now, the $r\bar r,g\bar g,b\bar b$ "bicolors of gluons" (only two combinations of the three are allowed) are actually closer to the photons than the mixed colors. So it's these bicolors that produce an attractive force of a very similar kind as photons – they are generators of the $U(1)^2$ "Cartan subalgebra" of the $SU(3)$ group and each $U(1)$ behaves like electromagnetism. That's why these components of gluons cause attraction between opposite-sign charges and repulsion between the like charges.
The six off-diagonal "bicolors of gluons" (and let me repeat that the actual formulae for the interactions never separate them from the rest – they're included in the same color-agreement-blind sum over color indices) cause neither attraction nor repulsion: they change the colors of the interacting quarks so the color labels of the initial and final states are different. It makes no sense to compare them, with the idea that only momentum changes, because that would be comparing apples and oranges (whether the force looks attractive or not depends on the relative phases of the amplitudes for the different color arrangements of the quarks).
At any rate, particles like protons contain quarks of colors that are "different from each other" so they're closer to the opposite-sign charges and one mostly gets attraction. However, the situation is more complicated than it is for the photons and electromagnetism because of the six off-diagonal components of the gluons; and because gluons are charged themselves so the theory including just them is nonlinear i.e. interacting.
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Flavor is entirely orthogonal to color - the gluon(s) neither "know" nor "care" about flavor, but the existence of different quarks still leads to phenomena descending from the strong force that you wouldn't get without them:
What remains of the strong force on the scale between nucleons is often called the residual strong force and can be thought of as being mediated by pions. The pions, being bound states of up- and down-quarks, evidently would not exist in a world with only a single quark.
Another phenomenon related to the strong force is that confinement depends on the number of flavors, too, see e.g. this answer.
So while on the level of the QCD Lagrangian there is nothing that would indicate at first sight that the different quark species and gluons interact in interesting ways (and indeed a theory with only a single quark species and gluons would be consistent!), there are emergent phenomena that depend on both gluons and different flavors.