Understanding the Zariski topology on $\mathbb{R}$

Yes, you have it exactly right.

What you're observing is that compared to the standard topology on $\mathbb{R}$ -- i.e., the one with a base given by open intervals -- the Zariski topology is very much coarser: every Zariski-closed set is standard-closed, but the converse does not hold.

In fact the Zariski topology on $\mathbb{R}$ -- or on any field $k$ -- is simply the cofinite topology: i.e., the nonempty open sets are those with finite complement. This is the coarsest topology which satisfies the $T_1$ separation axiom*: i.e., that singleton sets are closed. It is not a Hausdorff topology unless the field $k$ is finite....in which case it's the finest possible topology -- the discrete topology.

*: For several years now I have taken to calling such spaces "separated" (rather than "$T_1$", "Frechet-Urysohn"...). So far no one has objected...probably because no one has noticed.