From the links posted here, it sounds like they're adjusting how the attributes on the death records are counted. So, all the times the "had COVID" attribute was missing before, it's still missing now. That's the original undercount, and it's still there. So, no change to that first difference.
The links seem to say, in the past when the stat included COVID on the record it was included more "generously" than what they're changing too. If that's time, or other situation based, it could be lots of stuff. It feels like the original measure was trying to not miss anything while the new measure is trying to draw a more strict cutoff.
It looks like someone who caught COVID and it pushed some other condition over the top, or contributed to some other condition being worse, these are the likely differences. How they draw the line between COVID was the direct and primary cause, COVID was a significant contributing factor, COVID wasn't significant but it made some other condition appreciably worse, COVID wasn't significant but just an additional straw on the pile, to something more removed.
The impression is they're trying to shift where that line is drawn. From a public health and reporting standpoint, you have to draw the line somewhere. Most of these are not clean lines, otherwise most people would die from stopping breathing or their heart stopping. Everything else just contributed to one of those. From a single person personal standpoint, it doesn't matter if COVID is "the one" or just "the last straw". The personal impact is the same.
Most complex stats like this are fuzzy and not black and white. Which is also one of the reasons the excess deaths number is cited often. It doesn't try to call out "the one cause" but just that circumstances changed and the number is way higher the last few years. We then infer that COVID is the largest changed circumstance without trying to tell if it's direct or indirect. It's the bolder thrown in the pond even if the ripple on the edge is what is noticed.