Neoliberalism is closer to Milton Friedman economics than anything Obama advocates.
Obama is mostly a neo-Keynesian.
There's a truism within the field that "There are no schools of thought in Economics." In part this is just a cheap rhetorical ploy to spring on the initiated but it
is genuinely descriptive of the homogeneity of professional/academic maceoeconomics and the technocratic apparatus it's a part of. You can't properly speak of competing research programmes of neo-classicism, neo-keynsianism, new-keynsianism or whatever. Those terms are only indicative of broad historical trends within the field that over time become absorbed into the mainstream profession. Which isn't to say that disputes about, say, policy, don't happen within econ. They happen frequently and they're often heated, but the site of conflict is usually an issue that is completely alien to popular understandings of partisan allegiances (e.g., given how close we are to the zero lower bound, what's the proper NGDP we should be targeting). These arguments are also communicated in a rigidly formal mathematical language that I can 1000% assure you no politician would understand.
I thought of neoliberalism as more of a political ideology than a system of economics- specifically a general empowerment of capital beginning in the 70s, done through eroding labor's power, the deregulation and growth of finance, trade liberalization.
From what ive seen, it entirely depends on who's writing. This is pretty much the standard definition you'll find as its used pejoratively by the Naomi Klein types on the left and if I was allowed my own addendum I'd throw in "general animosity towards the welfare state and social services more broadly." Whenever you need your obligatory, generic neoliberal "what is it?" citation I see two sources most frequently, David Harveys
A Brief History of Neo-Liberalism and any number of works from Philip Mirowski, usually
The Road from Mont Pelerin. Within different subfields it varies widely: for Wacquant, a sociologist, the neoliberal historical turn coincides with a new need for states to police and incarcerate their populations; for James Ferguson, an anthropologist specialized in developmental Africa, it allows for state-legal apparatuses to cordon off non-sedentary peoples dependent on barter exchange into what essentially amounts to literal labor pools; for Brad DeLong, an economist who worked in the Clinton administration, and is a self professed "card-carrying neoliberal", neoliberalism is an admission that any honest take on the market needs to allow for regulatory policies and non-market solutions in order to preserve private property and enterprise (he also thinks neoliberalism admits of a hard, Thatcherite, version and a soft, Clintonian one).
It is very muddled. Not even the people who generally know what they're talking about agree on it. But I do think you're allowed to dismiss the more shrill characterizations that sling it around as an epithet to describe everything to the right of Bernie.