(Not sure if this should be on the Economics SE or Skeptics SE)
The first source is this, from which there is an image:
This clearly implies that the US federal budget decreased under Obama, Carter, and Clinton.
The second source is this, from which there is an image:
This seems to indicate that the first source is wrong, because the US federal deficit went in only one direction (up) under all three presidents that allegedly decreased the deficit.
How can these numbers be reconciled?
Debt and deficit are two different things. The deficit is the amount by which the national expenditure exceeded its revenue, each year. The debt is the total amount owed to other parties, accumulated over many years.
For example a country might start a year with 20 billion dollarydoos (DD) in its treasury accounts. If it then had a revenue of 100 billion DD while spending 130 billion DD, then for that year it had a deficit of 30 billion DD. It also now has a debt of 10 billion DD; it spent all of its existing treasury stockpile, plus all of the money it made that year, and therefore must have had to borrow an additional 10 billion DD to be able to spend more than both put together.
If next year it will have revenue of 110 billion DD and spend 120 billion DD, the deficit for next year will be 10 billion DD. The debt will accumulate to 20 billion DD; the country will have to borrow another 10 billion to cover next year's excess spending, and there was no spare money to pay down the original 10 billion debt so that is still owed as well.
At a very simplified level, as long as the annual deficit is non-zero the total debt will get larger every year. So even if the deficit is 5% smaller in one year than it was the previous year, the debt will still have increased in both years. Discussing a "change in the deficit" is talking about the trend of a trend. The debt will only decrease if the "change in deficit" is so large that the deficit becomes negative (a "negative deficit" is more traditionally known as a surplus); when there is more revenue than expenditure, then it can be used to pay down debt.
In addition, the first chart is not talking about the absolute value of the deficit, but the deficit expressed as a percentage of GDP (a rough measure of the total amount of economic activity in the country, not just government revenue or expenditure). So if the GDP is going up, the deficit will be a smaller percentage of it even if the deficit is actually the same absolute value, or even if it increases by a small enough amount. Thus in those years where the deficit as a percentage of GDP is shrinking, the actual deficit may in fact still be growing. And the debt will definitely be growing, while there is still a deficit to talk about at all.
(And of course in those years where the deficit as a percentage of GDP is growing, it may not indicate that those administrations were actually raising the deficit directly; it could be that they are just very good at lowering GDP instead)
The value "change in deficit, in percentage points of GDP" depends both on the absolute size of the deficit and the absolute size of the GDP, whereas your second set of data depends (and reports) only the absolute size of the deficit. The country's GDP is very likely to be different from the start of a term/administration to the end/next.
You can get get a negative value for this first chart's data by either (1) keeping GDP constant and decreasing the debt, or (2) keeping the debt constant and increasing the GDP. The latter because now the debt is a smaller fraction of the GDP than it was before, meaning you've reduced the fraction of debt to GDP. More generally, a combination of things can happen, and as long as GDP gains outstrip the debt gains you'll get a negative value of "change in deficit, in percentage points of GDP": you could double GDP and increase total debt by 1.5 times, and now your debt goes from an initial p% to .75*p%, which is relatively smaller, resulting in a net value of -.25*p% for the "change in deficit, in percentage points of GDP".
This seems to indicate that the first source is wrong, because the US federal deficit went in only one direction (up) under all three presidents that allegedly decreased the deficit.
Well, the correct way to [somewhat] relate the 2nd table to the first graph is to see that the debt went up less during (e.g.) Clinton's presidency than the rest you've mentioned. There's the confounding issue that the graph has full terms combined but the table is split by term. (Furthermore, as Ben pointed out as I was writing this, the first graph is also relative to the GDP, while the 2nd one isn't.) Still, you can see that 7% in Clinton's 2nd term is the lowest debt increase in that table, and that is compatible with the largest deficit reduction claimed in the 1st graph.
Comparison between those two metrics aside, whoever was president is clearly only part of the story in the 2nd table as e.g. "cohabitations" of a Republican congress and a Democratic president (both during Obama and Clinton) resulted in less debt increase than when either had a Democratic Congress. I suspect that if you had a bar graph for deficit changes split by term, you'd see that there as well.
There's one more thing to note, namely that the 1st graph is not (sorted) in historical order. As you can see from the table, "who came before" matters in terms of how big of bar you get, e.g. Clinton came after Bush the elder who saw large debt increases, so Clinton's deficit reduction (relative to Bush the elder) is also large (and negative) because of that. But G.W.Bush gets the largest increase in the graph because he came after Clinton, even if debt increase during his time doesn't look all that different from Bush the elder in % terms from the 2nd table (at least per term). And Obama gets an "about par" graph bar because he came after G.W.Bush, which is a non-obvious reason for the bar differences between the two of them in the 1st graph, at first glance. Essentially what I mean by this last para:
And if you're willing to forgo GDP relativization, but want to see the deficit (as opposed to the debt) evolving, something like this is probably more useful
(They also have [very large] table that also has a column ['chart' 3.D] with that relative to GDP, but no graph, and I'm feeling too lazy to make one myself.)
By the way, the NYT (1st graph) numbers were only projecting for Trump's term in 2017, so don't have Covid effects actually included.
Actually, here's the /GDP version of that with debt done the same (from their data). HTH
