44 622 / 700x45c what tyres to buy?
Some of the confusion relates to the fact that there are different but equivalent ways to measure tire sizes. The first tire measurement you mentioned 44 622 refers to the tire width of 44 mm and rim bead seat diameter of 622 mm respectively. The second measurement type (700x45c) is a 45 mm wide tire that also fits on a rim with a 622 mm bead seat diameter, the 700c is an old French sizing nomenclature based on the final external diameter being 700 mm for width class of tire (e.g., a, b, c widths). (The final outside diameter is affected by the widths, so the three widths would have different bead seat diameters in order to get a final diameter of 700 mm).

The differences in tire widths you are seeing (i.e., 42, 44, 45, 47) likely corresponds to the fact the final tire diameter depends on the combination of tire and rim widths (i.e., the same tire can have different final widths depending on the rim width it is mounted to). One standard may assume one rim width, while another standard likely assumes a different rim width.

In the end none of these nuances matter much as a frame that can clear a 700x42 tire will also likely clear a slightly wider tire (e.g., 700x47).  What really matters is that all of these must be fitted to a wheel with a bead seat diameter of 622 mm, which is the diameter used on your bike.