My younger son used to work for a water testing company and he explained what exactly it is that water companies call hardness.
Hardness is a measure of divalent metal ions, which are mainly calcium ions, with some magnesium ions and trace amounts of other metal ions. But they add all the ions together and express it as though it was all calcium (or calcium oxide or calcium carbonate). Although they use the unit mg/l Ca for hardness, it is not a measure of the amount of calcium in the water but rather what the total of all the divalent metal ions would be if they were all calcium.
Looking at your screen shot, you need the hardness figure to the right of the table. The figures in the table are the analysis of each individual metal, not the hardness.
So, ignore the figures in the table; look at 'hardness' on the right hand side of the page (108.9 mg/l calcium) If you click on the 'convert your hardness' button, and convert your 108.9 mg/l calcium into mg/l calcium carbonate (also called ppm) and German degrees, those are the two you need for fish keeping. I converted it to 15.5 deg and 272 ppm, you can check my maths
(My younger son now lives in Yorkshire, so I went to Yorkshire Water's website and used his postcode so I know the convert button works)
Then you need to go to
Seriously Fish and look up the fish on the list in your first post. Some profiles on there use ppm, some use German deg (
oH)
Black phantom tetra - 18 to 215 ppm
Dwarf cory (I looked at pygmy cory) - 2 to 15 deg
Rasbora (I looked at harlequin rasbora) - 18 to 215 ppm (search for Trigonostigma heteromorpha)
Cherry barb - 36 to 357 ppm
Dwarf rainbows - 5 to 15 deg
As you can see, your water is a bit too hard for black phantoms and harlequin rasboras; just about OK for pygmy cories and dwarf rainbows; and very suitable for cherry barbs.
It is much better to get fish that like your water than to try to alter the water to suit the fish you like. The latter can be done, but I would gain some experience before risking this option.
Fishless cycling takes a few weeks so you have plenty time to finalise your wish list. I would use this time to go round all your local shops and make a note of all the fish that catch your eye. Then go home and look them up on Seriously Fish, seeing which ones match your tap water hardness.
But be warned. The vast majority of shop workers don't believe in fishless cycling. If you tell them that's what you are doing, they'll either ridicule the idea or tell you to buy a bottled bacteria product and buy some fish at the same time. Resist!!!!!