I acknowledge that 'socialism' is a vague term with dozens of definitions, but this strange strictly-American idea that publicly-funded infrastructure is socialist isn't a useful definition, nor a common one. It will really just confuse people.

Historically and presently, socialism is a labour movement which, despite all the variations, had the common goal of the workers controlling their means of production, rather than the owning class. Almost every political dictionary and socialist will back that up, and also Wikipedia (for something we can check right now). It's not about whether something is private or public.

Paying taxes and voting in a (systematically broken, throroughly corrupted) government representative democracy isn't really accomplishing this. We are arill beholden to the owning capitalist class. How I spend my working hours is still governed by a bourgeois board of directors, I don't own the tools I use, I don't have meaningful power to make democratic decisions about my work or my society governance.

You are correct that socialism exists (present tense! see: Zapatistas) without planned economies. But if you want to see what socialist modes of organisation look like within capitalism, it would be a workers cooperative.

Anti-car movements are not socialist nor socialism. They are good and pro-society, but are completely incidental to the socialist movement.

Collectively-funded operations like roads, police and our military airstriking hospitals aren't socialist nor socialism. We have no control over the use of our money and labour; even if voting was democratic power in practice, a campaigning platform isn't a guarantee of policy, they can completely ignore that once elected. And also, no matter who you vote for, your tax money will still go towards anti-socialism!

As for the parts about communism, well, no. The definition you've invented wildly conflicts with both theory and historical events. You're gonna have to start from scratch on that one, even just looking at the Wiki article will provide a much better base. Very popular ideologies like anarcho-communism just completely contradict all that.