For some reason the Kaiweets is not in as close agreement with the Brymen as before. 
You'll see variations like that which is why even though you might often see closer agreement, you have to figure +/-1C as reasonable tolerance. Things just move around a bit.
I assume from what I'm seeing that your room air temperature is about 13.8C, give or take a bit? So what we're seeing is that the Siglent agrees with the Brymen as to the measured voltage and the short is effective enough, but there's a bit discrepancy between the internally measured cold junction temperature and the actual effective cold junction temp--the latter being a bit of a complex topic. In your cold room, the internal measurement is probably correct for where it is but then there is likely a large temperature gradient between that point and the actual terminals. I think this will be more pronounced in a cold environment and your results seem to indicate that the actual effective cold junction temp is about 14C. Try putting some insulating materials--cloth, bubble wrap, some masking tape--to cover and insulate the entire right front quarter of the meter including top and bottom. Seal it up as best you can and wait a few hours. I'll bet that if you put two probes in your cup tied together near the end, one to the Siglent and one to the Brymen, that after a few hours your readings will be much closer.