So, (according to a Reason article), Obama was speaking about unproven gas reserves. EIA now estimates that these will now last for 86 years.
The U.S. Energy Information Administration estimates in the Annual Energy Outlook 2023 that as of January 1, 2021, there were about 2,973 trillion cubic feet (Tcf) of technically recoverable resources (TRR) of dry natural gas in the United States. Assuming the same annual rate of U.S. dry natural gas production in 2021 of about 34.52 Tcf, the United States has enough dry natural gas to last about 86 years.
However, EIA also points out that these figures are highly uncertain.
Estimates of TRR are highly uncertain, particularly in places where relatively few wells have been drilled. Early estimates tend to vary and shift significantly over time as new geological information is gained through additional drilling, as long-term productivity is clarified for existing wells, and as the productivity of new wells increases with technology improvements and better management practices.
On the other hand, proven reserves were quite small in 2012 a well; the Reason piece said those were going to run out in 12 years (back then):
The U.S. currently consumes a bit more than 22 Tcf of natural gas per year producing electricity, heating buildings, and providing feedstock to chemical plants. Proven reserves are 273 Tcf which would be completely depleted at the current rate of consumption in about 12 years. So if known reserves are so small, is it delusional to think that the U.S. has a 100-year supply of natural gas? Perhaps not.
The Reason piece also notes that proven reserves were less than 10 years in the 1990s as well, yet by 2012
the country has cumulatively produced and burned twice the amount of proven 1996 reserves.
Wikipedia (citing EIA) gives more updated figures for US proven reserves to last 14 years as of 2021. One thing that is certain is that US proven reserves have significantly increased over the past decade (right figure).
Latest figures of production and near-term estimates thereof (some of which is exported as LNG--around 15%) are
EIA projected dry gas production will rise to 100.27 billion cubic feet per day (bcfd) in 2023 and 101.68 bcfd in 2024 from a record 98.09 bcfd in 2022.
Converting that to per-year gives around 37.11 Tcf, so somewhat of an increase compared to the 2021 estimate in the 1st quote (34.52 Tcf), which gives about 80 years estimate of TRR lasting, under the same simple EIA model. A comment below says that the long term [intended] substitution of oil with gas in US domestic consumption should be accounted for as well. The EIA model doesn't do that; they use a simple "flat" projection model for future production/consumption estimates.
On the other hand, even with the simple EIA model/data, production was only 22 Tfc/year in 2012 and it is projected slightly above 37 Tcf next year. (The TRR seems to have gone up as well.) So some gas production increase did happen since 2012. I think the 2nd Obama quote you gave is too vague (typical of politicians though), e.g. has no time frame, to make some factual determination as to what he was suggesting was going to happen in terms of switching from oil to gas.
Your (2012) Slate criticism piece said for instance that is was about switching to gas (just) from coal:
They also advocate switching a substantial part of our power generation from coal to gas, in order to reduce carbon emissions. Were we to do those things, that 21-year supply could quickly shrink to a 10-year supply, yet those same advocates never adjust their years of supply estimates accordingly.
It is also true that coal-based generation has fallen in the US since then.
So on that vague level, what Obama said was true: gas production went up, and coal went down. But I don't see any time frame in the Obama speech even for coal elimination. Perhaps if you're willing to consider the more recent goal setting under Biden's presidency that:
The United States remains the world’s third largest coal consumer — even after a decade that saw the country close a quarter of its coal capacity. Another quarter is scheduled to close by 2030, according to U.S. Energy Information Administration figures.
Perhaps someone estimated what that would require in terms of [increased] gas production by 2030, but I wasn't able to find some such estimates quickly. It's probably not a straightforward calculation since renewables might also partly substitute.
Also, oil was hardly used for electricity generation in the US:
There is perhaps more to be said about [directly] substituting gas for oil in the auto sector, but there are electric cars there to consider as well. It seems in the US those are limited to busses, and there don't seem to be plans/projections for much increase (e.g. there are about 160,000 such vehicles in the the US, compared to 1.8 million in Argentina.) Anyhow, if you're willing to consider all this, it's a complex model due to increase in electricity consumption owing to electric cars.
The EIA does also release some long-term energy use projections for the US. At least last year they projected that natural gas use would stay roughly the same by 2050, with renewable making up the difference for the reduction in coal use.
That's only the "reference case" pictured; the margin of error probably comes as additional models, with varied assumptions. FWTW, the 2023 version of that is even more optimistic, as is seems to predict a decrease in natural gas use in the "Ref" case, and even the uncertainty bands don't seem to exceed the current production [much] for gas.
On the other hand, despite the projected near constancy in US gas use for electricity production, they do predict a moderate increase in US gas extraction due to exports! Eyeballing their graph (since I'm too lazy to dive in their XLS files for the precise figures) it's projected somewhere around 41 Tcf for 2050, with a high case of 50 and a low of 30.
Whether any of those projections are realistic should probably be the topic of another Q, as we're getting a bit far afield from what Obama explicitly said in that speech. (Anyhow, these more detailed models only go till 2050, so have only a 30- rather than 80-year time frame. It might have been more suitable to use those projections from around the time Obama made his claim, but I'm not sure those can easily be found, without diving into archive.org. And would it have been reasonable for 2012 models to predict the increased LNG demand abroad due to the war in Ukraine?! Insert the saying about unknown unknowns here.)
As a parting thought as to the predictive power of such models, it turns out that the AEO2012 is easily found on-line (at least an "early release" thereof, if you search for the right keyword), but it predicted that by 2035 natural gas and coal shares would have both been rather constant, even by 2035.
Of course, we now know that's hardly what actually happened in the past decade. However, the AEO2013 through AEO2015 were basically the same story, with nearly flat-line predictions. It was only in AEO2016 that some dramatic change was forecasted, and that was rather contingent on the Clean Power Plan [CPP], although even the flat line coal/gas/renewables prediction was down to 26%/35%/23% by 2040. But in the CPP alternative, those shares were 18%/38%/27%, i.e. they were roughly hoping half of the coal displaced (3-4pp) was going to be due to gas and the other half due to renewables. The forecasts that predict only renewables will replace coal seem to be of more recent introduction. (I haven't look exactly when they started to make those.)