Political words are notoriously slippery. The project of defining them can sometimes feel like stapling Jello to the wall. This is especially true of words that develop strong positive or negative connotations. Connotation can attach itself to the meaning of words if the connotative use becomes common enough.
The word "socialism" has taken on a life of its own over the last few months. The variety of uses of the term are making political dialogue terribly confused. Average Americans honestly discussing their political differences will often find themselves arguing over what does or does not qualify as "socialism." This is because words are weapons, and both sides have an interest in dulling their opponents blades and sharpening their own.
I'm here to help. I have done my best to parse the common uses of the term "socialism" and subdivide them into their proper categories. Keep in mind that these categories will only reflect term usage in the aggregate. Individuals tend to equivocate.
The first thing to notice is a broad categorical distinction between "socialism" referring to a historical thing and "socialism" describing modern political feelings. There is debate within both categories, of course. But let's begin with the historical.
Properly defined, historical socialism includes fascism (national socialism) and communism (international socialism), both of which sought to empower the working class by giving it control over the means of production. National socialism gave workers control through nationalization of industry (accomplished either by state regulation of nominally private corporations or by direct state ownership). Communist socialism sought to empower workers by overthrowing the state on behalf of a global proletariat—a project which, ironically, always necessitates tyrannical use of state power.
Democrats unanimously desire to exclude the national socialists from historical "socialism," for good reason. The farther one goes toward the political Left, the fewer historically socialist regimes actually count as "socialist." For Democrats, there are countless ways to parse history based on which socialist regimes you want to identify with, and which ones you want to throw under the bus. For some, Soviet communism was a form of socialism that was too radical. Out on the fringe, the Soviets were not radical enough to be "true socialists."
For a number of historical reasons too complicated to get into, here, the word "socialism" developed a negative connotation in American politics in the 20th century. Because of this, modern Republicans had a political interest in broadening the term, using it to describe Social Security, Medicare, ObamaCare, and many other federal programs. Republicans hoped to smear some of the negative connotation of the word onto Democrat policies. And it worked! But, in doing so, Republicans gave birth to the modern, colloquial usage.
The way younger generations use the term "socialism" is determined by a memory that only extends over the last few decades. Millennials have largely been taught that history is a story of diverse people-groups standing up to oppression. And, in this story, America has no special place, and is more often than not the oppressor. What little they learn of the history of socialism is seen through this lens: socialism, being on the side of the people, is on the right side of history.
These Millennials have grown up listening to Republicans refer to basic social programs dismissively as "socialist policies." They have also heard Republicans defend income inequality on the basis of merit, which, to them, seems little different than a blanket assertion that successful races have more merit than less successful ones. For Millennials who lean to the Left, "socialism" became a reactionary badge of honor much like the word "deplorable" did for Trump supporters. The term more precisely describes what they are against (Reagan-era meritocracy) than what they are for.
The best definition I can come up with for this new "socialism" (often called "democratic socialism") that is sweeping the Democratic Party is this: "the prioritization of equal economic outcomes among social groups over meritocracy." Conservatives who want to be able to speak intelligibly to Millennials need to understand this.