No, Ameria is not losing at globalization, and no, the deal about it hasn't changed from the 90s to the 00s. Look into any macroeconomics 101 textbook written over the last 50 years, and you will find the following consequences of globalization described clearly and convincingly:
1) In terms of productivity, and of the material wealth that comes from it, free trade benefits the US, as well as any other nation participating in it.
2) In terms of income distribution within nations, free trade will benefit those factors of production that are more abundant nationally than internationally. (In the US, that would be capital, education, and natural resources.) This will come at the expense of the factors that are scarcer nationally than it is internationally. (In the US, that means labor.)
3) For the scarcer factor (labor in the US), the balance of the two effects can could go either way. Hence, if free trade's effect on productivity dominates, American workers win on net; if its effect on distribution that dominates, they lose on net.
With this in mind, it's perfectly predictable that free trade would boost the incomes of Texas oil barons, Silicon Valley programmers, and capitalists in general. It's also predictable that the auto workers in your article are stagnating, perhaps even falling behind.
But just as the effects of globalization are described in two generations of economics, so is the solution. It isn't to opt out of globalization and forgo the benefits of international division of labor. It's to compensate workers for their distributional losses. Today, an obvious candidate is to introduce universal healthcare. Other compensations might include the tax system. Personally, I would expand the Earned Income Tax Credit.
So to conclude -- no, it's not the competition with other countries America is losing at. It's loosing at mediating the inequities at home. And instead of showing some civil courage and stand up for some liberalism at home, the press blames the rest of the world for America's self-inflicted wounds.