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The marking on the existing tyre to replace is 44 622 /700x45c
On sites like https://www.schwalbe.com/en/groessenbezeichnung.html 44 622 corresponds to 700x42c and 47 622 corresponds to 700x45c , I am confused which number to trust and 44 622 also looks like rare tyre size as I can't seem to find any in UK.
Also how strict are these numbers +-1 size jump going to render me unable to put tyre on?
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.
You should be fine.
A long time ago in France, wheels were measured by the diameter in mm over the tire and you had size 700, 650, all the way down to 400 on kids bike.
There were also different widths, so they went 700A through 700D from narrowest to widest.
This system is great if you are trying to figure out if your fenders are going to fit around the tire, but terrible if you are trying to match tires against rims, because a skinny 700A is going to not be very tall and thus have a larger diameter rim than the balloony 700D. And rim diameter is what matters most for tire compatibility.
Over time most of the other sizes went extinct except for the 700C and 650B. Other countries all had their own ways of measuring.
The real way to tell if it matches is to measure the diameter where the tire meets the rim. That is your 622 number.
Most shops will be calling it by the 700C, not the 622, but if you get them is will say 700C in ink, but 622 in the raised letters. Don't worry. That rim diameter is extremely popular.
The smaller number is the tire width. That's less important and everyone measures it a bitt differently.
Schwalbe do something a bit odd, as you've spotted. The widths of the same tyre don't match for the different numbering systems for some of their tyres. I believe this is the ones with thick, stiff layers of picture protection under the tread, which would make them take a different shape to a simpler tyre, further from a perfect torus.
Unless you're extremely tight for clearance, it doesn't matter. If you are that tight, the same nominal size from a different brand or even style would need a test fit. Example: I've run anything from 700x28 to studded 700x38 on my hybrid; the later needed some fiddling with the mudguard to fit.
