By "interaction with PRODUCT*V1" do you mean the effect of PRODUCT when VISIT=V1? If so, then this comparison is known as a "simple effect". It is not interaction.
An interaction would be implied when the effect of PRODUCT when VISIT=V1 is different than the effect of PRODUCT when VISIT=V2. Perhaps this is what you mean by "global interaction".
"Meaningful" is not the same thing as "significant". The effect can be meaningful (in context) without enough evidence for (statistical) significance if a p-value is large-ish (which some people might interpret as p > 0.05) but sample size is small-ish (implying a potential lack of adequate power). The effect can be (statistically) significant without being meaningful (in context) if the sample size is large-ish and the p-value is small-ish.
It is possible for the "global" interaction test to have a large-ish p-value while one of the simple effects components has a p-value that is small-ish. My interpretation in that scenario generally is that there is too much uncertainty to conclude that the interaction is "significant". But both "meaningful" and "significant" are context-specific, and there are many nuances.
The distinction between meaningful and significant, and the role of a p-value in statistical decisions, has an extensive and argumentative literature. The recent special issue of The American Statistician https://www.tandfonline.com/doi/full/10.1080/00031305.2019.1583913 is a good place to start.
I hope this helps.
