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.
It is important to realize that the partial pressures of dissolved gases will eventually reach equilibrium with the surrounding atmosphere; the lungs are a great gas exchange organ (that's their entire job), so blood leaving the lungs should have partial pressures that approximate those inspired air (in practice, there is some discrepancy because the lungs are very humid and also are constantly refilled with $CO_2$ from blood, so water vapor and carbon dioxide contribute a substantial gas pressure that pushes out other gases: see the similarity between arterial blood and alveolar gas in your data table and the differences between atmospheric gas and alveolar gas).
Clinically/in a lab, we measure these things with a machine that just magically gives the numbers. I looked for a simple description of how these machines actually function, and found https://acutecaretesting.org/en/articles/understanding-the-principles-behind-blood-gas-sensor-technology to be useful. In summary, $CO_2$is measured by exposing a captive solution to the gas and measuring the pH, giving an indirect (but accurate) measure of $CO_2$. $O_2$ is measured with a reducing current. It's also possible to measure concentrations of arbitrary compounds more directly with gas chromatography - colleagues of mine have used this for anesthetic gases, for example.
