Well, in general you are trying to minimize delta E.
But delta E is comprised of different components, and the color control (usually, mostly) affects the luma component, so if you are only changing color, improving delta luma will probably improve overall delta E. If that's not the case on your display, then yes, the larger goal is to reduce delta E, so don't get carried away chasing low delta luma values.
To review, the Delta E row shows total error. The two rows below that attempt to show the two sub-components of that error. Delta xy is a measure of how far off the hue and/or saturation are. Delta luma isolates just the "luma" (luminance, intensity, whatever- not sure Color HCFR terminology is Poynton-compliant here) error.
Stepping back even further, the CIE chart shown in HCFR is actually a view looking "down" on a 3-dimensional volume of valid colors in xyY space. This top-down view shows x and y, but not Y. The Y dimension comes up out of the diagram towards your eye.
There is a good visualization of this space here:http://www.brucelindbloom.com/ChromaticityGamuts.html
So to get back to delta xy, that has to do with where you are in a given xy "plane" sliced through the xyY gamut. It's easy to see the dot move in the xy plane on the CIE chart in HCFR. The further it is away from where you want to be, the greater the "delta xy".
But it doesn't tell you whether that reading is too intense or not. That's what the color control affects, and that's what is being addressed by the delta luma row in HCFR.
When you're relatively close in xy-space then the delta luma readings can be relevant.
(The way HCFR calculates delta luma assumes you are very close to the standard primaries. If you want to know how far off you are from an ideal luma when you are forced to use primaries that are in non-standard locations in the xy plane, you'll need to use a spreadsheet. Your primaries are close enough in my opinion for the default delta luma calculations to have meaning.)
On most displays, the "Color" control mostly changes the "luma" of the primaries all at once. There may be changes in saturation (how much you are displaced inward or outward along a radius extending out from the white point) or hue (how much you are displaced clockwise or counterclockwise along a line *perpendicular* to a radius extending out from the white point) as well, but mostly this control affects how far above or below the ideal height above the xy plane you are. This is not easily visualized in the CIE chart in HCFR, so the delta luma row can be a useful aid in setting color correctly.
(It's also a good way to expose differences in luminance between the primaries. Sometimes displays are deliberately designed to "push" red to be more intense than green or blue for instance)
So to bring it all back to minimizing delta E, you would like to be able to control all three dimensions in the xyY space. Color and Tint are crude controls that affect all primaries and secondaries at once. Color mostly changes Y position, tint mostly changes xy position.
You say you don't have a tint control, so all I'm focusing on is the color control.
If two or three of your primaries are too intense, turning down the color should lower *overall* delta E. (How you decide what is the best way to determine "overall" delta E is, especially when a change moves the delta E for different colors in opposite directions, is a little more complex, but usually you can tell if you're improving things or not.)
If adjusting the color control raises delta E, then your luma is either close enough already or your display's color control is affecting hue and saturation as well, which would show up in the delta xy row.
I couldn't actually read the last jpegs your posted - the resolution was too low. As annoying as it is when people post massive jpeg's, there is such a thing as being too conservative. Of course you can always just post a .zip file containing all the relevant .chc files output from HCFR (the title of the thread even encourages it after all).