Nice bit of parsing and justification by Vox to say precisely nothing. Let's look at the numbers from the last year of the Bush admin to 2015 (the last year the numbers are available on the website). Pay special attention to the Returns column.
Year Removals Returns
2015 333,341 129,122
2014 407,075 163,245
2013 434,015 178,691
2012 416,324 230,360
2011 386,020 322,098
2010 381,738 474,195
2009 391,341 582,596
2008 359,795 811,263
Removals have stayed relatively constant with mostly a 75k variation from a low point in Bush's last year until Obama set a new low in 2015. Still, by and large, deportations have fluctuated between around 385 and 435k.
Returns, on the other hand, fell off precipitously from 811k in Bush's last year to 583k in Obama's first year. Now, in fairness, most of that can probably be attributed to economic circumstances at the time. Yet it continues to fall to new lows every single year until it hits 129k in 2015. This is an extraordinary drop when you consider that the Bush and Clinton years were seeing numbers in the millions under this column. You simply can't chalk it all up to a shitty economy, nor can you chalk it up to the new removal rule, which did not see a corresponding increase in numbers similar to the decrease in numbers here. And in spite of the new removal policy carrying more official consequences for repeat offenders, it's not like it's a deterrent to illegal immigration. They still take their chances and hope they don't get caught. The new policy simply gives ICE and BP more teeth and the ability to detect patterns in removals.
No, the obvious explanation here is the new policy was a nice cover to point to when somebody criticized your enforcement policies as lax. "But look, I've removed more people than any other President!" Lies, damned lies, and statistics is all that is. His overall policy has been enormously forgiving, even prior to DACA (which also explains some of the drop).