Your "regular spindle" is one that I think is no longer made. It is a 33 Jacobs taper with locking collar and requires a special chuck which, if still made, is hard to find and will be Chinese. In the picture, you can see the collar, which has a hole for a pin spanner wrench, just above the chuck body. These spindles were common around 1955-1980 and had a real advantage. Normally, a standard 33 Jacobs taper chuck is extremely difficult to remove once installed on the spindle. The locking collar version was made to be easily removed. The reason for removing the chuck was that there were several accessories that could be installed on the drill spindle in place of the chuck to allow mounting Morse taper drills, buffing, grinding, routing and shaping wood. Those accessories are quite scarce now.
The alternative spindle design that you call "extended" has a 2 Morse taper that allows using either Morse taper shank drills or a chuck with a Morse taper arbor. The extended length is to allow for the extractor slot to be accessible.
A few decades ago, Rockwell would sell some of their 1/2" capacity drill presses with either the 33 JTC or a 2 Morse taper spindle. It is quite easy to change the spindle from one to the other, if you have the parts. My 17" Rockwell variable speed drill came with the 33 JTC spindle. I made a new 2 MT spindle for it when I rebuilt it because I want the flexibility of being able to use Morse taper shank tools, either drill bits or different types of chuck.
The picture shows a Rockwell 17" quill with the 33 JTC spindle installed. Note the large male thread above the 33 JT. The collar on the chuck serves to pull the chuck tight onto the taper and also to force the chuck off the taper. Below is a 2 MT spindle, which is a pretty simple part to make from a 2 MT drill extension. As you can see, the part of the spindle that goes into the lower quill bearings is too slender to allow the 2 MT taper to fit inside the bearings, forcing the entire 2 MT portion to be below the bearings. Larger drill presses sometimes have the Morse taper up inside the quill bearings. In those machines, the extractor slot is only accessible when the quill is all the way out, exposing a slot through the quill to allow the wedge access to the slot in the spindle.
Some cheap drill presses will come from the factory with a cheap drill chuck already installed on a spindle with a Jacobs taper or even a European taper. It will probably be a difficult job to remove the chuck without damaging the spindle. And it may be difficult to find a good chuck to replace the bad one these days.
I advise getting a drill press with a Morse taper spindle unless you are looking for a smaller drill press with 1/4" or less capacity.
Larry