Electrical – Higher brightness LED with bigger resistor to save battery life

I think the problem is a lack of understanding of what one means by "brightness". Brightness is a very ambiguous term. First of all, I'll make the assumption that you are speaking in terms of a human's perception of brightness. This limits our discussion to photometry. You have to now differentiate between two measures of brightness, i.e. Luminous intensity and Luminous flux (or Luminous Power). Luminous flux is the light power emitted in all directions and has SI units of "Lumens". Luminous Intensity is the light power emitted in one solid angle which has SI units of "candela". Think of a solid angle as a cone with it's vertex at the led emitter. You mentioned mcd as units, these are millicandela, so mcd is a measure of Luminous Intensity. You can't directly compare luminous Intensity (candela)) with Luminous Flux (lumens). I did find this tool to convert between the two units, but, it must have to make the assumption that the light is omnidirectional and is only accurate for such lights.

You'll have to get your units straight before you can make sense out of this.

In additon, you have to decide what your objective is. Are you trying to make a light with high luminous Flux or one with high Luminous Intensity? I would imagine that you are going for a light with high luminous Flux or you probably would be speaking of laser diodes instead of LEDs.

So choose whatever package that maximizes your Luminous Flux, i.e. Lumens. You did not provide enough information to give a better recommendation than this. That is, you did not specify any additional design constraints.

While the answers by @Passerby and @MichaelKaras pretty much cover it, there's one more thing to add:

1. Humans perceive light intensity non-linearly: At very low intensities, we are very sensitive to a even slight variation in brightness. On the other hand, at higher intensity, the unassisted human eye is pretty much incapable of discerning differences in intensity.

   This really interesting graph demonstrates this excellently: _^(source)

   Essentially the ability to perceive change in intensity is very high when most of the vision is attributable to the rods of the eye (scotopic vision), and drops very low when the cones are doing the sensing (photopic vision), i.e. at slightly higher luminance.

2. Less critical but good to know: LEDs illuminate somewhat non-linearly versus current, with the graph dropping off the linear as current increases. This is most noticeable with red.

   _^(source)

So, long story short:

The human eye cannot notice even large intensity changes at the higher levels of light that an LED generates at higher currents. Using half or even less than half of nominal rated current (20 mA typical for indicator LEDs, 50mA or more in high power LEDs) will thus work perfectly fine for most indication purposes.

In my designs, 5 mA is my preferred current for all indicator LEDs: Try it, it works great!