The use of terminology differs across disciplines. I'm answering as a political scientist.
Your study would only be a (quasi-)experimental design if there were some aspect of randomization in assignment to treatment. Based on the information you've provided, you are dealing with an observational study because the units self-assigned to treatment. This means that there are measurable and unmeasurable features of the observations that differ systematically between units in G1 and G2. In a school setting, there are essential features of students who go to state schools and those who go to private schools. Ideally (but probably not practically) you could control for these differences. I am skeptical because there are probably unmeasurable differences between the populations, such as motivation, family support and upbringing. These are things you cannot control for.
The key feature of any experiment is random assignment to treatment. If units are selected at random to receive treatment and then all units adhere to their assignment, then in expectation all treatment units and control units will be similar in their pre-treatment characteristics. Then the researcher can credibly infer that any post-treatment differences are due to the effect of treatment.
An "experiment" involves an intervention by a researcher that randomly assigns treatment. In my line of work, random selection is only possible in a very limited range of research designs. As a result, I do not consider it a requirement for a "good" or "true" experiment in my field.
A "natural experiment" involves randomized assignment to treatment through an intervention executed by something (human or not) other than the researcher.
In my experience, "quasi-experiment" refers to a research design where there is some aspect of randomness to assignment to treatment but not to a degree where the researcher believes that treatment and control units are similar in expectation in terms of pre-treatment features. This term, however, is quite vague and subject to different meanings. In my field people advise against its use. I've heard some claim it's too vague to be useful, obscuring more than it reveals about the research design. I've also heard some claim that "quasi-experiment" is just a euphemism for "bad experiment." I agree with the former sentiment and tend to disagree with the latter.
An "observational study" involves no randomized assignment to treatment. In these settings a researcher must try to establish "conditional ignorability." The term means that after controlling for a set of covariates $X$, a unit's potential outcomes are conditionally independent from assignment to treatment. In other words, the treatment assignment and the potential outcomes are conditionally independent given $X$ if and only if, given knowledge that $X=x$, knowledge of whether $T=\{0,1\}$ provides no information on the likelihood of a particular outcome in $Y$.
That is, conditional on the covariates, the treatment is independent of the potential outcomes.
\begin{equation}
(Y(T=1),Y(T=0))⊥T|X
\end{equation}
A "correlational study" would probably not be of much value. A "correlational analysis," however, probably means that you present exploratory data analysis that shows the unadjusted relationship between variables. In other words, without any claim of causality, you are showing how variables tend to move together in your data set.
The formalization above is based on the Rubin potential outcomes framework for causal inference, a foundational work for this conversation. You need to read it if you're serious about moving forward with work like this.
Holland, Paul W. 1986. “Statistics and Causal Inference.” Journal
of the American Statistical Association. 81(396): 945-960.
Another approach to conditional ignorability is Judea Pearl's back-door criterion.
