At first read I see a flaw in the first part of your argument, which is that centralization vs collectivization of economic ownership is not directly indicative of policy, rather it is the percentage of economic output which is used for collective services that dictates the Left/Right spectrum, which indeed is how a Far Left position can be coextant with a market economy and private ownership but with a tax or public stake in economic actors that returns a majority of the “profit” to collective service, it is rather the degree of enforcement of property rights as one of a set of rights and regulations by a central Authority which lies on the Auth/Lib spectrum that dictates the structure of the economic order. This is how for example you could be a Lib/Left Marxist who prefers central planning of the economy, so long as you don’t believe the central planning should be enforced by monopoly of violence and instead implemented by collective consensus, there is no fundamental conflict in the position. Leninism on the other hand implies use of force by a centralized state military/police to restructure the economy along central planned lines, which is an Authoritarian position.

I agree the “quiz” is very flawed, it would need an order of magnitude more questions to be accurate, and authorship bias is certainly an issue.

That said, the compass itself I find to be quite accurate to the mental political models of most individuals. What you are pointing to, Centralization vs Distribution, is a relatively new way to concieve of the older Federal vs Local or State vs Community political framework. I would indeed view this as a “third axis” or omission by the two axis compass, as both Authority and Economy can have organization and flow biased towards fewer or more numerous nodes of participation/enforcement. To go back to your Lib Left Marxism, you could say that the Marxism part of that formula calls for a State economic planning model with high collectivization of economic output and low State enforcement of policy. On thing often missing from the Auth/Lib axis description is that reduced State enforcement does not mean reduced enforcement overall, but rather that the enforcement does not rely on the state monopoly on violence, instead directing enforcement through social exchange relying on the individuals applying their independent power onto each other to discourage deviancy from the consensus.

An easy example of this is in many tribal groups and including pacifist Western religious sects the worst corrective action an individual faces is shunning, which relies on all of the individuals of the community independently choosing to no longer participate socially or economically with the individual being corrected. The decision to do so may be more or less centralized or decentralized (for example a Priarch/Priest might declare shunning in a nonviolent Christian community, while a specific tribal group may only do so through a process of full group consensus, or even the most lib/local of all a spontaneous reaction of each individual against the deviant based on norms.