Horse manure is considered a hot source of Nitrogen because it is fairly readily available, however composting it changes that. The Nitrogen in compost is not very readily available but it is there, just not very soluble and your test looks for soluble Nitrogen. The same with the N in leaves, it is there just not very soluble so tests do not show much.
The test I have had done on my compost, most often made with horse manure and shredded leaves, show an N content in the 1 to 2 percent range, but the plants growing in soil amended with that compost grow quite well. The soil bacteria release the Nitrogen as the plants need it.

It is your tester that is at fault :-) Nitrogen is the most mobile and unstable of all the soil nutrients and unless using rather sophisticated labs and apparatus, virtually impossible to measure to any degree of accuracy. Certainly with any home testing kit. Heck, even many professional soil labs do not test for nitrogen because its levels can vary from day to day, hour to hour and even in sun versus shade so they are unable to report with any accuracy!

Any reasonably decent organic matter that is not wood based - compost, manures, leaves, grass clippings, etc. - should provide very adequate levels of N for at least one growing season.

What is the LaMotte actually measuring? Ammonium, nitrate...? Just curious. And how is it calibrated or checked?

good question. I have no clue so researched. This is what I found when google LaMotte soil nitrogen
"This Nitrate Nitrogen Test Kit tests for nitrate nitrogen levels using cadmium reduction which undergoes diazotization/coupling to form a pink color. Levels are determined using the Octa-Slide 2 comparator. Range and sensitivity is 0.25, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0 ppm NO3--N."

I have absolutely no idea what it means. Shades of pink I could figure out.

I read elsewhere that Nitrogen is really hard to test for so decided that I shouldn't worry about it. Don't seem to have problems associated with nitrogen deficiency so am cool with that.

Thanks everyone.

It is very easy to add too much Nitrogen too, which then would promote lush green growth that would be more attractive to insect pests and plant diseases. Excess N can also interfere with the plants ability to utilize both major and minor nutrients.