I don't know if the light reflections are this bad (What would you say?) but I can change the position of my camera to remove those. Could a lightbox improve the quality of my photos? Other thoughts are welcome!
If the purpose of these photgraphs is a catalog, don't change anything: they're close to perfect! I say this as a professional photographer.
Reflections are not "per se" bad. In your case above, the reflection shows nicely how the surface is.
There is no single "correct" way to take product photographs. As a photographer you should play around with the light and the effects of the light. Sometimes a specific reflection is bad, sometimes it is just right. But you (photographer) should have an idea what you want to show.
Regarding the light box, yes a light box will distribute the light more equally. It will inevitably make your products appear more "flat". If your products do have high contrasts, or even have glittering metal surfaces, this may be the best choice. But I don't see this for your shots above.
There's also a significantly cheaper alternative to a light box: Transparent paper, like vellum paper, HERMA Glassine or Kraft paper. If you put a big sheet of this paper in front of your light, you can "modulate" the dispersion of the light with the distance from the light to the paper.