Looking at your photos again since you made them larger, it is very clear that the heat damage is not the result of the soldering but is what looks like a sustained overloading of the power supply and I bet that is the HT side that your having the issues with. Those caps also look like the originals and so I'd tend to treat those as suspects as well but I'd also look for excessive current draw on the plug in cards as well and that could be shorted tants and other caps on the power rails.
That tester looks to be OK, heres mine, the XJW01 is the one that is most most accurate and displays plenty of information as you can see and you can select the test frequency of 100Hz, 1KHz or 7.8KHz and it will automatically detect what it is your testing and go to that range, ie., L, C or R and their is also a calibration setting where you can further refine the accuracy. People have tested these against much more expensive HP LCR meters costing many times the price and the results have been very close indeed.
As you can see from the photos, there is a only a small difference in the readings between the mains and the battery unit, both were testing the same cap, a 1000uF 50VDC and some of the difference may be down to the testing frequency the battery one, I have no idea what frequency it tests at and the XJW01 should really have set at 100Hz for that test as it was cap typically used for power smoothing so would be seeing most likely a 100Hz rectified frequency.