Neither of those is "socialism." Socialism has a strict definition.

From Wikipedia:

Socialism is a political philosophy and movement encompassing a wide range of economic and social systems which are characterised by social ownership of the means of production, as opposed to private ownership.

Socialism doesn't necessarily say we should remove all forms of private ownership, as much as it says we should remove private ownership of the means of production. Also, that doesn't mean that nobody owns it, it means all the workers own it, collectively. It doesn't necessarily mean that you can't have your own house or your own refrigerator, rather that the companies that build those will just be collectively owned. Imagine every company being unionized by default, something like that.

We have a similar structure of owning things in the US when it comes to stocks and public companies, but the thing is in that case anyone can buy a part of the company. In socialism, only workers who are invested in the company through their labor get part in ownership and choice of the direction of the company.

I do agree on higher taxes for the rich, but that's just a band-aid on the existing capitalist system, it's not really a "socialist" idea at all. Higher taxes on the rich is about trying to keep capitalism from spinning out of control into outright feudalism. The rich only become that rich to begin with because of how our system of ownership works where they take all the excess profit generated by their workers and keep it for themselves. Socialism would remove their ability to do that because they would own as much of the company as any worker. They would no longer hold dictatorial control on the finances and where they go.