I'm writing a report and need to be precise but concise in the abstract. Currently I called it 'multiplicative multiple linear regression'. But when I Googled it, not much came up. In the same vein, I called multiple linear regression without the interaction term, additive multiple linear regression. But what is the official way to call it and be concise about it?
Technically, it is still called 'multiple linear regression' (assuming you do, in fact, have multiple predictors). I have at times seen the term 'multiple polynomial linear regression' when polynomial terms are added, but this is not that common; I've never seen 'multiplicative multiple linear regression'. In your abstract, you might consider simply noting that you included interaction terms. Better yet, if your field allows it, reference 'eq. 1', whereby in text, the full, expanded model with all terms are spelled out.
You can use the term moderation. For example, let's say your dependent variable is some sort of self-esteem measure and you want to see whether the effect of some intervention on self-esteem is moderated by (depends on) gender, then your regression model has self-esteem as the dependent variable, and intervention group, gender, and the intervention*gender interaction term as predictors.