I have found that timing is everything for reusing yeast. I have a primary which is a minipubs plastic 23 ltr fermentor with sediment traps, and a secondary glass carboy. I keep both full at all times. When its time to brew a new batch, that is the same time to transfer secondary contents to ball lock keg. I swirl the yeast sediment up from the carboy, with remaining beer and pour it into a sterilized ( with sulphite) one liter mason jar. There is no trub. I get about 1/2 jar. Then I transfer the primary contents into the cleaned and sterilized carboy. I dont get much trub in my primary; and I cant stand to pour that yeasty stuff down the drain, so I swirl that up and pour it into the mason jar until it is full. I seal and shake, and start my new brew. In a few hours I pitch that full mason jar into the new batch.
So, brewday is the day where everything is done at once, all transfers and yeast preparations included.
The yeast is never rinsed or the liquid replaced with water. I WANT beer to be the liquid to be introduced into the new batch. The new brew is chugging out the fermentation lock within hours, sometimes immediatley, only rarely does it over foam out the top. I also ferment at 45f , so the yeast is always on the cool side of its temperature range. I think this helps the process but I'm not sure. I never have to store yeast separately, its always right there working away, and goes into the next brew with no idle time in between. I mainly use lager yeast as my fermentation room is at a constant 45f most of the year (fall winter spring).. If I make ale, buy a new strain, make a starter, and run off a bunch of ales, for the warmer months when the temps are in range. I live in Innisfil, ontario, so an insulated garage with a thermostat controlled radiant heater keeps temps where they need to be except one to two summer months. I use Saflager 34/70 primarily. This system works for me and pumps out incredible beer at the rate of 10 imp gallons per month.