As a notation note, I never seen your $P_a{O_2}$ notation used except for referring to arterial partial pressure ($P_v{O_2}$ would be venous). Generically, partial pressures are usually noted as, for example, $P_{O_2}$.
When we talk about partial pressures in biology, we literally mean the partial pressures: that caused by dissolved gas.
Hemoglobin-bound oxygen is not the same as gaseous oxygen, nor is carbonic acid the same as carbon dioxide. Neither contribute to the partial pressure. They do relate to the total oxygen/carbon dioxide carrying capacity of blood, but this cannot be measured directly through partial pressures.
See this Q&A for a situation this comes up: Why is arterial pO2 normal in carbon monoxide poisoning?
In medicine, it is common to use oxygen saturation as an alternative measure of blood oxygen concentration; this refers to the percentage of oxygen binding sites of Hb that are saturated.