Checking with vendors' offerings and with the very knowledgeable salespeople at my work, I could find no particular "name" for a necklace like you described. Various styles of necklaces do incorporate a clasp as part of the decorative lower part but these are not what you are looking for. These are the "lariat" styles in which one end of the chain is longer than the other, fitting through the shorter end.

My suggestion for a front mounted clasp and pendant:

Any competent jeweler should be able to modify a chain to do like you want. The catch to the clasp situation is having an attractive clasp arrangement so all works together to look nice on the neck. If looks are not important, then any sturdy clasp will work, such as a 'lobster claw' or a sturdy 'spring ring' clasp.

One alternative to the regular clasp styles is the folded metal wire 'hook style' clasp. The wire is bent into a loop so the open end is slightly against the other side, making for a neat friction fit where the pendant loop is slid into. An example of this style is on the following website. Scroll down and on the right side you will see 'NBd109 Aquamarine nugget' in a description. Look at the clasp. This is a simple folded wire hook arrangement.

Using this style of clasp, a pendant may be permanently attached to one end of the same style (with the hook soldered shut) and then the other side may be slid onto the pendant loop to fasten the chain together.

Using Spring Ring and Lobster Claw Clasps: These are the typical clasps used in necklaces. The lobster claw style is superior in strength.

For clasp like this to work properly, the pendant will need a loop going the correct direction. Imagine that the cross pendant is placed flat on a table surface. The loop on the pendant should lay flat also. In other words, the front view of the pendant will show the 'looking through the hole' view of the loop on the pendant.

One side of the chain is attached permanently to the loop (left flexible, of course, attached with another loop of metal) and the clasp will be attached to the loop. With the loop on the pendant facing as I described, all will hang properly. Obviously, if the loop on the pendant faces the other way, like it would to simply slip over a chain, the pendant will be pulled sideways when the clasp is attached.

As an alternative to changing the loop already on a pendant, an additional loop may be formed and soldered shut, going through the pendant loop. This will give the pendant more 'movement' when worn and will take care of the direction problems.

Ask a jeweler: As I said before, any jeweler competent enough to solder sterling silver should easily be able to take a chain of almost any style and modify it to have a 'front' closing clasp which on one way or the other attaches to the loop(or loops) of the pendant. The attractiveness of the overall piece will depend on the clasp used.

It is possible to use two clasps of a decorative style with one on each side of the pendant loop. The hook style clasps work quite well visually but are not as secure as spring loaded types like the lobster claw. Still, the hook style might be best for appearance sake. These solutions will keep the clasp and the pendant in front without the problem of 'roaming clasps' that wind up somewhere on the collar bone or so when worn.