Your definitions of capitalist and socialist are inverted. A socialist state pays for things from tax revenue. Capitalism means the individual pays directly for a service in the market - the state is not part of the transaction.
Take education for example. A socialist education system is when the state runs it and people get it for "free". A capitalist education system would be a free market in education where the individual pays directly for the service.
That's a textbook definition. If you think anyone in the real world obeys the text book definitions to the letter, you would be wrong.
Corporations today are pushing communism very hard. They've been getting tax subsidies up the ass for so long in systems that cal themselves either, why wouldn't they?
Capitalism means no state involvement in the economy. That's the definition of the word in a textbook or a real world conversation.
Corporations are a legal entity created by the state so companies have limited liability. (If the corporation pollutes a river for example then the individuals who are responsible are not financially liable.) Corporations are not capitalist, despite what the leftists on reddit will tell you. They will erroneously define a capitalist as a greedy money grubber, but in truth it means someone who wants liberty from state control. You can see why a leftist would want to conflate freedom with greed.
Corporations lobbying the government for favorable legislation is also not capitalism. Again, it's the state interfering with the market economy. This type of activity is better described as mercantilism or crony capitalism.
To review, if the state is involved it's not capitalism.
What we've been seeing in the west since ~1900 and before is a steady movement toward more socialist governments. Since the definition of capitalism is always misrepresented in the media people often blame capitalism for socialist policies and the problems that come from it.
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