I find it somewhat hard to imagine the 3 blue "petals" as the foreground. I'm having trouble working out which one is in front of which one, and that keeps niggling and messing with my belief that they might form a consistent whole. However, the blue circle is easy to see as a background. It makes perfect sense on its own, if you remove the foreground. And the lines make perfect sense as a foreground, so there's no reason to see them as a background.
This is reversed for Lauren Ipsum's flower: each shape makes sense as a petal of a flower, but the blue shapes taken together don't make much sense as a set of gaps between lines, whereas the lines are simpler to describe as gaps than as lines. The lines could be a foreground, but they don't quite line up where they expect them to. If they're a background, they make perfect sense.
I can see two basic ways to make your blue regions seem like a foreground. First, you could use a different colour for the parts where two circles overlap. This would create a visible interaction between the three circular regions, so that it was clear they were regions. It would also preserve the symmetry of the logo. But you would have to be careful to make sure the 4 inner regions don't become the foreground.
Secondly, you could make the white lines seem more like a background. I think any pattern of lines will make some sense as a foreground. But I think it's possible for the lines to make more sense as a background than as a foreground. One way to do this would be to make each of the intersection regions, between each pair of petals, slightly smaller. Then each long edge of such a region would not quite line up with the corresponding edge of the centre region. Also, the thickness of each line would vary in a specific way. It would be easy to describe the varying thicknesses and not-quite-lining-ups as the gaps between the seven blue regions. But if you tried to describe the way the lines changed thickness, it would be difficult. This would make the white lines seem more like a background image, because it would be simpler to conceptualise them as gaps than as lines.
Lastly, you could change the border of your logo so it isn't a perfect circle. The arc describing the outside of each outer region doesn't have to be centred at the centre of the circle. You could draw any other circle through the two corners of the region, and you would still get a symmetrical image if you made the same change to all three outer regions. This would make the overall blue shape make less sense as a background.
