The HDD size being 2.5" or 1.8" has nothing to do with it fitting without it bulging the keyboard. The HDD is a 2.5", but Dell has gone for a 7mm height HDD. Most SSDs are 9.5mm, some even taller. Some SSD
that will fit and not bulge, but without the HDD case, for example those that are between 8mm and 9.3mm.
Strangely I see that the video OP is linking to and some others here have used Crucial m4 but not the HDD caddy. The Crucial m4 can be converted to a 7mm (Crucial did this on purpose). You can take the top off, remove the plastic spacer in between, and then put the top on again and and voila it's a 7mm and can fit in the caddy. Also Intel sell 7mm SSDs alongside the others. Some other brands also allow you to convert easily.
Look here for Crucial m4 conversion
storagereview dot com / crucial_m4_ssd_review_256gb