It seems to me that the question is more about "Why does karet is described differently in different puskim?", rather than "are there different kinds of karet" (and a possible answer could be "since there are different kinds of karet"). Especially, when considering other karet descriptions, such as in Shmot 12:16:
כִּי כָּל-אֹכֵל מַחְמֶצֶת, וְנִכְרְתָה הַנֶּפֶשׁ הַהִוא מֵעֲדַת
יִשְׂרָאֵל בַּגֵּר, וּבְאֶזְרַח הָאָרֶץ
and in Vayikra 7:27:
כָּל-נֶפֶשׁ, אֲשֶׁר-תֹּאכַל כָּל-דָּם וְנִכְרְתָה הַנֶּפֶשׁ הַהִוא,
מֵעַמֶּיהָ
From pshat point of view, there are no different types of karet, and all different wordings have the same meaning. The people of Israel is sometimes called collective קהל, sometimes עדה and sometimes plain Israel (there are, of course, nuances between the words but they don't change meaning in the macro).
What is exactly that meaning is not consented. In Moed Katan 28a, there are two explanations. Raba explains:
אמר רבה מחמשים ועד ששים שנה זו היא מיתת כרת
and Abaye says that Raba's karet is called "years" karet (כרת דשני) whereas there's also "days" karet (כרת דיומי - Rashi explains it's sudden death מיתה חטופה):
רב יוסף כי הוה בר שיתין עבד להו יומא טבא לרבנן אמר נפקי לי מכרת אמר
ליה אביי נהי דנפק ליה מר מכרת דשני מכרת דיומי מי נפיק מר
Rashi in Kritot 2a (dibur hamatchil "פסח ומילה מצות עשה"), however, says that karet is dying without children:
וכרת הוא שמת בלא בנים
And if you further read the informative hebrew wiki page of כרת, you can find additional different commentaries for karet, none of which refer to the difference you indicated in the question (מתוך הקהל/מישראל).
I is used as drash in Horayot 9b. the specific word הקהל in B`Midbar 19:20 is used in a halachich drash to explain R' Shimon's psak that Kohen-Mashiach doesn't bring a korban (sacrifice) even if he entered the mikdash while impure (and for some reason the Gemara/Rashi/Tosfot don't ask how chachamim interpret that word).
