Your first interpretation is closer to the truth, at least on average.
But the actual truth is pretty complicated, and sometimes you can reliably run at some Absolute Maximum Ratings without any issues.
There is no generic answer, and this is because manufacturers use the same language to mean different things. Some manufacturers give "recommended" operating conditions. For some, absolute maximums might mean some "reduced lifetime" ratings, but this is not a standard practice. For others, those ratings might mean that non-destruction is guaranteed, but operation according to specs (like current consumption, noise, etc.) is not. This is a moot point, too, because all the other specifications are defined in some specific operating point anyway (like Vcc=5.0V, Ta=25degC).
If you are having a fight over this, it's a telltale sign that neither party fully understands the problem. Of course, admitting your limitations in understanding, and erring on the side of caution when uncertain, is your best bet.