In this case, deep poverty is defined as households with income under 50 percent of the poverty line (about $10,000 for a family of three). The calculation is based on more accurate measures of poverty that have since been endorsed by the Census Bureau.
Now, this is a different measure of poverty than the one used by Kathryn Edin and Luke Shaefer that I noted yesterday. Their measure is both tighter (looking at even lower poverty rates) and looser (it counts households that are in extreme poverty even for short times). So it's not entirely an apples-to-apples comparison. Still, once you look at the historical numbers, it doesn't look like the 1996 welfare reform act slowed down the growth of welfare spending, nor did it have more than a very small effect on deep poverty.
None of this is especially meant to defend welfare reform. But 20 years later, it doesn't look like it really had quite the catastrophic impact that a lot of people were afraid of at the time.