Let me stress first that the fine-structure constant cannot be calculated by theory; it is a fundamental constant of nature and we only know its value through measurements. Many people, most notably Pauli, have been puzzled by the value of the fine structure constant (it has been said that Pauli died in room 137 in one of Zurich's hospitals).
The way $\alpha$ is extracted from measurements depends on the type of measurement and typically involves calculating the observed effect with an initial ''guess'' for the fine-structure constant. The value of $\alpha$ is then changed untill the calculation accurately predicts the observed effect which can be the Combton wavelength, Lamb shift, hyperfine splitting etc. It is important to realize that in some of these methods the theory is assumed to be exact. These different experiments probe different effects where $\alpha$ plays a role and require different levels of theory. In extracting the fine-structure constant from spectroscopy of hydrogen and muonic hydrogen for instance, the theory is limited by how accurately the "size" of the proton is known. In the experiments involving positronium it may be that higher order QED effects have to be taken into account that have been neglected in prediciting the observed decay times.
Every four years, the Committee on Data for Science and Technology (CODATA) issues recommended values of the fundamental physical constants and uses the best determinations of the fine-structure constant to derive the recomended value of $\alpha$, which isbasically a weighted average of the different data points.