Making your own lathe using a hand drill as the power source is a great way to get started in lathe work. But there are some deficiencies. Paul explains...

And if you are interested in more lathe stuff I have opened up a whole new section of the website here: Lathe Projects

One of the reasons I moved up to a dedicated lathe was the 'Non' power end. That is the center.

On this type you cannot change the center. The B&D I showed and the one in your link does not turn with the mandrel. The point merely locates in a dimple/tiny cavity in the end of the mandrel and is a friction fit. It can be overtightened and in turn overheat the mandrel. It is not ideal.

The better ones have a live center, meaning that it revolves due to having a bearing plus it can often have four sharp edges to engage with a wooden project such as a turned leg of a chair. In pen turning the live center still locates in the mandrel's end cavity but turns as the motor turns, avoiding overheating but if over tightened can bow the shaft.  

Even better is the mandrel saver which not only has a bearing it allows the mandrel to slide into a hollow tube thus saving overheating and the bowing associated with overtightening when encountering a blank which rotates freely and wont allow the tool to function as application stops it rather than cutting it. In my case, I found it was the glue failing to hold in the blank. Re gluing cured it and I could finish my turning. NB just make sure the Morse taper is the correct one for your choice of lathe. Below examples of mandrel saver and how it is used.

Just thought I should clarify things a little bit better. In fact maybe I should have outlined the things a beginner might look out for when choosing a lathe and accessories required for starting turning?

Live Mandrel Saver

Grizzly H2669 Hobby Lathe/Disc Sander -

Mount your electric drill in this Hobby Lathe/Disc Sander and you'll have the ability to turn small items, such as pens and then sand them to completion! Includes a spur center, screw center, tailstock and toolrest for wood turning. For disc sanding, includes a 5" sanding disc that mounts in the drill chuck and a sanding table that mounts in the toolrest holder. Sanding disc uses any 5" PSA sandpaper discs. Electric drill and 5" PSA sanding discs not included. Features: 360Degree head swivel 6" swing over bed 24" between centers Aluminum alloy bed construction Can be set up as a wood lathe or a disc sander Requires drill with 44mm collar size. Weight: 10 lbs.