A Christian Socialist would simply be a Christian who subscribes to some form of Socialism. Socialism is a broad category term. Socialist ideas have been floating around in the West since at least the 18th century, though most people today tend to associate Socialism with the philosophy of Karl Marx–Marx though advocated for a very specific kind of Socialism that would lead to Communism, which he envisioned as a state-less and class-less society.
Marxism is a form of Socialism, much in the same way an apple is a kind of fruit, but not all fruits are apples.
I don’t know if there is a once size fits all “Christian Socialism” to speak of, though Christian Socialists have been around since the 1800s, advocating for a kind of Christian Socialism–a Socialism based upon and rooted in the ethical and moral teachings of Christianity. One of the more famous examples in American history would be Francis Bellamy, a Baptist minister and the author of the American Pledge of Allegiance.
Due to the Cold War and Red Scare, much of the West has tended to view any kind or even the mention of Socialism as having an association with the Soviet Union (and to later totalitarian governments such as Maoist China, North Korea, etc); and as the United States specifically saw itself in an interventionist way to put down Communist revolutions backed by the Soviets, the language and rhetoric has tended toward an absolutist good vs evil framing. This makes more nuanced conversations more difficult; but it also means that criticism of Capitalism is often viewed as anti-democratic and associated with 20th century demagoguery.
I’m not a Socialist, just to be clear. But I do believe that Capitalism, unto itself, is dangerously wicked; not because Capitalism advocates for personal private ownership, but because without strong regulation from the government, and without strong moral scruples of individuals involved, Capitalism becomes an untamed monster of corruption resulting in a lot of the problems we can see in the modern world: vast wealth disparity, the erosion of the middle class, dark money in politics, oligarchy, plutocracy, etc. Issues many Christian social reformers of the 19th and early 20th century recognized, hence the strong support labor reform, unions, and antitrust laws among Christian social reformers of previous generations.
My Christian faith prompts me to believe that of central concern, as it pertains to economics, is that of economic justice. I would therefore doubly condemn Capitalism as little more than a form of Social Darwinism; and would regard “pure” forms of Socialism (aka Marxism) as unrealistic utopianism that ignores the serious reality of sinful human nature: if you create a power vacuum, then the unscrupulous who desire power will naturally come to fill that space–and that’s how we get people like Stalin, Mao, Kim Il Sung.
So if our chief concern is, as it pertains to a healthy society here our neighbors can flourish, social and economic justice; then we should work to create systems which maximize human flourishing. And I believe that this a major component of what the Christian vocation of citizen and neighbor entails: recognizing the social and moral dimension to our Christian life, and working for the general betterment of our neighbor, especially the poor, the hungry, the widow, the orphan, and the stranger.
We therefore cannot reduce everything down to a Socialism vs Capitalism, wherein Socialism evil and Capitalism good. Because that’s not engaging in Christian ethics, that’s merely parroting certain 20th century political talking points without critical thinking.
If we desire to see good happen in the world, then the source we should rely on isn’t secular systems (Socialism, Capitalism, etc) but our faith in Christ. It should emerge as a byproduct of our Christian ministry and vocation as lovers of our neighbor. This isn’t “Social Gospel”, but it is an organic fruit-bearing of what it means to believe the Gospel and take the Gospel seriously in all of its ramifications.