Li-ion would be good if you find you regularly need more than three hours of light in a night. Otherwise, you won't need Li-ion. The nickel metal hydride battery will do fine.
I wouldn't weigh the total life of either sort of battery much in my decision, if I were you. I'd be surprised to see a NiMH battery lasting much beyond three years of regular use. I replaced the first NiMH battery in my L&M light after about 2.5 years of regular use. That was too long, too. I made do with short lighting cycles for a few months when I should have gone ahead and just replaced the battery.
I do use the light a lot, but even with fewer uses I would be surprised to see a NiMH battery lasting much beyond three years. So, the life per charge and the replacement cost of each sort of battery are the more relevant considerations.
If you do decide to use the Li-ion, there are two things you can do to lengthen its life. Store it in the refrigerator, or even the freezer, but warm it up before you use it. Also, store it much less than fully charged. You'll still get good life without doing these things, but you can stretch things out a bit with this sort of maintenance.
I would think you'll find that it makes a great deal of sense to replace the battery when it wears down. I'm still using the original light unit, after about four years, and it's going strong.