Everyone in the non-monogamous community has their own interpretations of the specific terminology, but I’ll offer my understanding of the various terms that apply to non-monogamous relationships.
Polyamory: I interpret polyamory to be having multiple relationships at the same time, be involved with a partner with multiple relationships, or be open to having multiple relationships. Polyamorists may have one “primary” relationship, or may have multiple relationships of approximately equal standing.
Open: “Open” is often an all-encompassing term for non-monogamy, though I have my own more specific interpretation for what it may mean as a label. To me, “open” relationships are primary relationships, where the members are allowed to date, but not have “serious” secondary relationships.
Swinging: Though there are various interpretations of the label among swingers, “swinging” generally refers to couples who engage in partner swapping and/or frequently go to sex parties and/or have sex, but not have emotional relationships, with many others.