Almost, but not quite. The letter of the code (the "current city-wide rule") is pretty clear that it permits no more than four unrelated people to occupy a single dwelling unit, regardless of how many dwelling units may be included on a given property.
There are two issues here. First is the relative looseness of fit between the intent of a law and the mechanism it uses to accomplish that intent. The intent seems clearly ("For purposes of controlling residential density") to set overall limits based on type (< 8 person houses, < 12 person houses, etc.), but to do that it sets a per-dwelling-unit number as the legal standard to enforce. The mechanism to accomplish that intent is to limit occupancy of a dwelling unit to no more than four unrelated people.
The second issue is the slippage between what is permitted and what you can get away with. The folks in the 6 person unit are actually non-compliant, though it could be that they could get away with it if the neighbors were not bothered by noise, etc. enough to notice and report them (and if review weren't triggered by some other factor related to density, like someone noticing that six people were trying to register cars using the same address). Since 4 per dwelling unit is the letter of the law, the fact that there was a "gap" of two people in another unit to keep the property under the intended cap would not be legally helpful.
So, in your final example, you would not be correct. The intent is indeed to regulate overall density, but the mechanism used is to regulate the per-dwelling-unit occupancy. The 6 unrelated people are non-compliant because they are unrelated. If they were related (a married couple and both pairs of their parents, for example) no limit would apply, but there are still no more than four unrelated people permitted in a single dwelling unit.
RE: Plus or minus four people
There are two issues here. First is the relative looseness of fit between the intent of a law and the mechanism it uses to accomplish that intent. The intent seems clearly ("For purposes of controlling residential density") to set overall limits based on type (< 8 person houses, < 12 person houses, etc.), but to do that it sets a per-dwelling-unit number as the legal standard to enforce. The mechanism to accomplish that intent is to limit occupancy of a dwelling unit to no more than four unrelated people.
The second issue is the slippage between what is permitted and what you can get away with. The folks in the 6 person unit are actually non-compliant, though it could be that they could get away with it if the neighbors were not bothered by noise, etc. enough to notice and report them (and if review weren't triggered by some other factor related to density, like someone noticing that six people were trying to register cars using the same address). Since 4 per dwelling unit is the letter of the law, the fact that there was a "gap" of two people in another unit to keep the property under the intended cap would not be legally helpful.
So, in your final example, you would not be correct. The intent is indeed to regulate overall density, but the mechanism used is to regulate the per-dwelling-unit occupancy. The 6 unrelated people are non-compliant because they are unrelated. If they were related (a married couple and both pairs of their parents, for example) no limit would apply, but there are still no more than four unrelated people permitted in a single dwelling unit.