So, while you're 100% correct about neoliberalism not belonging to either the left or the right, your basic description of neoliberalism isn't correct. What you describe (deregulation, positive valuation of wealth generation, free markets, etc) is just liberal capitalism.
Neoliberalism names the extension of market-based rationalities into putatively non-market realms of life. Meaning, neoliberalism is at play when people deploy cost/benefit, investment/return, or other market-based logics when analysing options, making decisions, or trying to understand aspects of life that aren't properly markets, such as politics, morality/ethics, self-care, religion, culture, etc.
A concrete example is when people describe or rationalize self-care as a way to prepare for the workweek. Yoga, in this example, becomes of an embodiment of neoliberalism: taking part in yoga is rationalized as an investment in self that results in greater productivity.
Another example: how it seems that most every public policy decision is evaluated in terms of its economic viability, and if it isn't economically viable (in terms of profit/benefit exceeding cost/investment) then it is deemed a bad policy. This is a market rationality being applied to realms of life that didn't used to be beholden to market rationalities.
Hence the "neo" in "neoliberalism" is about employing the logics of liberalism (liberal capitalism, I should say) into new spheres of life.
A good (re)source for this would be Foucault's Birth of Biopolitics lectures, which trace the shift from Liberalism to Neoliberalism. As well, there's excellent literature coming out of anthropology about neoliberalism at work in new spheres, in particular yoga, which is why I used it as my example here.