Kon, were you not alive or not old enough to remember Clinton's presidency? I grew up in the 80s and 90s, and I remember Clinton running a budget surplus for his last term or so, and that he left office with a budget surplus, I remember the media talking about it rather a lot, I remember the dot com bubble bursting (or the media's blabbering about it, anyhow), and I definitely remember 9/11, and what America was like before it, and the budget surplus turning into a deficit after President Bush got his hands on the ship of state, and the deficits skyrocketing due to Bush's wars and deficit spending on them, on tax cuts, etc. I also remember the sudden Republican opposition to deficit spending as soon as Bush had left office. It wasn't an issue for them before, but after 2008, it was a violation of their most closely held principles! We have always been at war with Eastasia Deficit Spending.
Considering that you've found a government website whose raw numbers have the appearance of disagreeing with history, I would posit that rather than those numbers being wrong, it is more likely that you are misunderstanding how they are supposed to be read (for example, perhaps the deficit as normally reported may not include interest, whereas the debt listed on that government website almost certainly includes debt accumulated as a result of interest payments).