I changed the oil on my HRX for 7 years not paying any attention to the amount of oil coming out of the engine, but I did always run it first for 5-10 minutes before turning it over to drain. If you have the ability to, you can also pull oh so gently on the pull cord while it is on it's side just enough to get the piston to move a bit and a small amount of additional oil will come out. I thought that the oil capacity was 18 oz but it would only take 10-12 to refill after a change?
Either way, bertsmobile1 is correct in that you should always check the oil with the dip stick just resting on the top of the filler tube and NOT screwed in. I would simply add a couple of ounces and check, and if it needed more, add a bit and check again. The entire time to do an oil change on these machines with warming up is less than 10 minutes so take your time getting the level just right. I have never used a pump, but they can certainly be beneficial for engines were the crankcase simply won't allow all of the oil to come out and with "suction" power it might help pick up extra deposits sitting on the bottom of the engine. Heck, tip and use a pump!!
Seeing as these engines do not have filters, I would typically run, drain, fill, run, drain and fill. This would help to flush out any deposits that would not come out with the first draining. I would also use Genuine Honda SAE30 since I wouldn't mower when outside air temps were under 50 degrees. If you mow in temps cooler than that, use their 10w30. 7 years later, the oil still came out clean after every oil change and I would typically average 25-35 hours per year of use.