Social safety nets are socialism. Go read any of the early Socialist writers and that is sometimes the only thing they really concern themselves with. Robert Owens comes to mind as an example. If you are using power to establish a "common welfare" then you are creating resources that are socially held corporations meant to be publicly accessed based on need. Basically it looks at entire populations fronting the resources and sharing in the security created from those services.

Capitalism at it's most basic is the idea of investment capital. Capital is the money made and returned in profit by owning a means to production, paying workers for their time or labour and scalping off the top of that. It expanded the idea of stockholders and privately held property. We as a society have done a very good job of memeifying capitalism into a bogeyman that is an all consuming monster... But the reality of the situation is that while capitalism has a nasty history a lot of it is tied up in how liberal veiws of everything as potential personal property and veiwing social ownership as essentially just unclaimed property... Isn't actually a core principle of capitalism. Capitalism can exist by degrees just as Socialism does. Both are ideas about what constitutes a legitimate ownership of property and production of goods but inside those ideas are a bunch of smaller concepts that are potentially compatible mixes that can co-exist.

But the other half of this is how are those labels currently being used in the discussion we as a society are having now. Pure strains of Liberal Capitalism are currently ruining the world because it equates freedom with unrestricted protection of privately held properties.

Social Capitalism is an alternative to Liberal Capitalism by a degree based on the works of socialists that look at what constitutes a free society based less on individually held property rights and instead a dynamic interplay of the socially and privately held.

If strains of socialist philosophical discussion holds strains of capitalism as a force naturally existing and not a problem in some form it is not a dichotomy, it's a parallel. Calling Market Socialism something like "responsible Capitalism" is something that is legitimately done by people who just don't like the connotations of the label "Socialist". You could scoff at them sure but people's personal emotional reaction to how they label themselves is still kind of relevant. If a Market Socialist and a "Responsible Capitalist" are advocating for the exact same thing then are they actually fundamentally different in any way that matters?

It all really depends on how you want to use the term. Ask yourself if this rejection is because you feel upset because your feel like your personal label is being attacked? If so where is that feeling actually coming from?