Why are different kaddishes said by different people?

Question

The "Mourners' Kaddish" and the "Rabbis' Kaddish" are said by mourners: either all instances are said by all mourners in the synagogue simultaneously, or each mourner gets a turn at one kaddish, or some other custom. but they are said by mourners, if present, regardless of who is leading the service.

"Half-Kaddish" and "Full-Kaddish", on the other hand, are said by only the prayer leader, irrespective of whether he is a mourner.

Why? What is it about the different kaddishes that causes this difference in who recites them?

Answer 1

Originally, there were 7 kaddishes for tefila,

3 during the morning service:

1. Following pesukei d'zimra
2. Following shemona esre
3. Following uva letzion

2 during the afternoon service:

1. Following ashrei
2. Following tachanun

2 during the evening service:

1. Preceding shemona esre
2. Following shemona esre

This was based on the verse "I praised you 7 times a day" (Tehillim 119:164) (Aruch haShulchan OC 55:4 based on Rokeach)

Later 3 more kaddishes were added after Aleinu. This was to allow mourners who were unable to lead the service to recite kaddish. Susequently, an additional kaddish was added after the yom (Aruch haShulchan)

It seems the original 7 were the purview of the leader, whereas the other kaddishes were allotted to the mourners.

Additionally we say a kaddish after the tehilla of chanukas habayis ledavid which (I recall learning but have to remember where) was added for additional mourners at the time when universally only 1 mourner said kaddish.

The kaddish derabannan may be said by anyone in the group that has learned torah shebaal pe, there is no leader. The custom is to let the mourner to say it, but even a non-mourner can say it.

Answer 2

There are different forms of kaddish which serve different functions.

Wikipedia correctly states, “In the liturgy different versions of the Kaddish are used functionally as separators between sections of the service.” Those are Half-Kaddish and Full-Kaddish.

This site gives a slightly more detailed explanation of the same point and says that the “Mourner’s Kaddish,” was specially allocated for mourners and placed after independent portions of the prayers, (most importantly after “Oleinu” - my addition).

ADDITION after Daniel's comment below: The site quoted adds that "The “Learner’s Kaddish,” or kaddish derabanan (Heb. קדיש דרבנן) is recited after a section of the prayer service that include an excerpt from Rabbinic teachings. When there is a mourner in the synagogue, he would say this kaddish too. When there is not, the leader of the prayers should say it.