I think you have shown that the drill is OK, I think it's very good, which is good news.
You have also shown that inexpensive chucks are pretty poor and all three are poor. You need to get a much better quality chuck if you need the precision
Out of interest, I did a check on my lathe with a chinese chuck on an MT2 taper.
At the lathe spindle, 0.0025mm runout.
At the MT2 arbor going in to the chuck, 0.005mm
At the chuck with various bits, 0.1mm.
On a 10cm ground steel bar, measured 23cm from spindle measurement point
Reading 1: 0.24mm, then release bar, rotate and tighten a number of times.
reading 2: 0.34mm
reading 3: 0.18mm
reading 4: 0.24mm
Not as bad as your 1mm, but I suspect that's what you will get with a 'non precision' chuck. Some may be a lot better.
I don't see any spec for accuracy in the Axminster user manual, so you don't know if it's in spec. I suspect it will be within their spec for the spindle.
They do specify the runout for their engineering series
"""The TB-16 bench pillar drill is truly designed for engineering and production use. Very heavily built and manufactured to exceptionally accurate tolerances,.This drill press will stand up to hard work day in day out. The headstock has been machined to precisely match the quill housing, virtually eliminating any spindle run-out, even at full quill extension. The tolerance of the spindle runout (Quill movement) is + 0.01mm. """ I don't see a figure for the runout at the chuck.
NOTE: your spindle is well within tolerance for their Engineering series, so you should be really pleased about that!!
I think the issue is the chucks. You either need to get Axminister to give you a better chuck or buy/borrow one to get the improvement you want.
and as
@Chief Sawdust Maker just said, make sure the mating surfaces are clean, page 13 in the manual.