I'm not positive, but I think the blend mode shown in the image is a "Value" (aka Brightness) preserving blend, based on the HSV color space.
I think the Luminosity blend mode in C2 is a "Lightness" preserving blend, based on the HSL color space.
HSL & HSV
HSL and HSV are similar, but ...
In HSL, setting "Lightness" (L) to 100% will turn any color pure White, regardless of what it was to begin with. That's likely why the Luminosity blend didn't recreate the effect you were looking for.
In HSV setting the "Value" (V) to 100% will turn pure Black to pure White and will turn any color to the brightest possible version of that color which preserves the RGB ratios, thus preserving hue.
I've used a lot of image processing tools and in my experience a "Value" based blend is a pretty uncommon thing which is why C2 likely doesn't have one built in. That said, it can definitely be added thanks to C2's extensibility.
Here is an example in which I "Value" blend white rectangles into the same image you provided.
The results appear to be essentially identical.
What's going on
Because the white square has a "Value" of 100%, it turns black to white, and boosts the colored areas such that for every pixel, the largest RGB component is maxed and the other components are scaled up proportionally preserving the original hue.
This makes sense if you think about how it affects black. For black, the RGB values are all the same (0,0,0), so the maximum among them is "0". If you slide the "max" value of 0 up to 100%, and then slide the other values up (also zeros) to maintain the same ratio of RGB amounts, then all the values will now be 100%.
Likewise, for a color like a dark orange ( 50%, 25%, 0% ), setting it's "Value" to 100%, will make it a bright orange. The max among the RGB components is the 50% red. If you slide that 50% up to 100% (maxing it out) we have doubled it, and so to preserve the relative RGB ratios we double the other values as well. So we go from dark orange ( 50%, 25%, 0% ), to bright orange ( 100%, 50%, 0% ), and voila, we've maxed out the brightness ("Value") without changing the RGB ratios.
A side effect of the HSV formula is that colors will never get blown out, or clipped, the way they will if you use an addition-based formula.
Seeing it in HSV color space
Finally, here I've taken your original image and broken it into individual channels in the HSV color space.
As you can see, the "Value" channel has your white square simply pasted directly into the channel data, perfectly overwriting everything it overlaps with the "HSV Value" of white, which is 100%. However, the hue and saturation data from the background image are preserved.
So in summary. I think its "Value" blend.