if you pitch the liquid and not the "sediment" into your fermenter, i can't imagine you'll get much benefit (or fermentation)...that tiny cake in your jar has all the goodies in it.
the yeast have all dropped out of suspension and are sleeping on the bottom of those jars. if you only add the liquid and not the "sediment" (some of which is yeast), you'll just be adding some very diluted, stale, and old wort to a fresh batch of brew. yum!
when you're washing: if you dump the diluted slurry from your fermenter into, say, a one gallon jug, let it sit for twenty minutes or so, and then decant the suspended yeast into your jars, you'll get a much cleaner (i.e. less sediment) harvest.