The distinction between experiment and quasi-experiment is often made in undergraduate psychology courses.
In general, the term quasi-experiment is used to identify studies that fall short of a controlled experiment in some respect:
e.g., pre-existing groups; allocation of participants to groups in a non-random way. In some cases, repeated measures experiments are labelled as quasi-experimental especially if they confound order with condition, especially simple pre-post designs.
The purpose of the term "quasi-experimental" is to highlight that there are potential confounds to causal inference that would not be present in a controlled experiment (e.g., time effects, pre-existing group differences, etc.).
However, more generally, an experiment typically refers to doing something in the world and observing the effect. In this sense, experiments are also sometimes distinguished from purely observational studies.
So in general, researchers should generally not use term "experiment" to describe an observational study. However, "quasi-experiments" can be labelled experiments.
So to come back to your question: how do you know whether a study is an experiment or a quasi-experiment? You have to read about the design and see whether the conditions for a controlled experiment are satisfied.
In general, as you progress in your psychology education, you may find that you use more nuanced descriptions of studies than simply "quasi-experimental". For example, you might describe a study as a simple pre-post repeated measures design. Or you might say that it compared a treatment and control group using pre-existing groups. The point is that these descriptions are more informative than simply saying "quasi-experimental".
In terms of reporting procedures for quasi-experiments in APA style, I think as above, this question does not entirely make sense. There are a variety of different types of quasi-experiments. For each type, there would be common things to report. For example, in simply describing your design, you will say what are between and within subjects factors and how participants were allocated or not allocated to groups.
You might find these APA reporting standards useful: "Reporting Standards for Research in Psychology" https://www.apa.org/pubs/authors/jars.pdf
This APA Task Force article also describes many of these issues quite clearly around how to report methods:
Wilkinson (1999) https://www.apa.org/pubs/journals/releases/amp-54-8-594.pdf
