I think a lit gloss on the first sense is sufficient.
I don't think 楔を刺す and 釘を刺す are always interchangeable.
楔を差す is in chujiten and prog.

@@ -5 +5 @@
-<keb>くさびを刺す</keb>
+<keb>楔を刺す</keb>
@@ -8 +8,9 @@
-<keb>楔を刺す</keb>
+<keb>楔を差す</keb>
+</k_ele>
+<k_ele>
+<keb>くさびを刺す</keb>
+<ke_inf>&sK;</ke_inf>
+</k_ele>
+<k_ele>
+<keb>くさびを差す</keb>
+<ke_inf>&sK;</ke_inf>
@@ -16 +23,0 @@
-<xref type="see" seq="1871830">釘を刺す</xref>
@@ -19,8 +26,2 @@
-<gloss>to make sure of</gloss>
-</sense>
-<sense>
-<pos>&exp;</pos>
-<pos>&v5s;</pos>
-<xref type="see" seq="2834064">楔を打ち込む</xref>
-<misc>&rare;</misc>
-<gloss g_type="lit">to drive a wedge (into, between)</gloss>
+<gloss>to make sure (of)</gloss>
+<gloss g_type="lit">to drive in a wedge</gloss>

くさびを刺す	0      not [uk] (book search, twitter confirms)
楔を刺す	33	0.3%   This expression
釘を刺す	11144	99.7%  Related figurative expression (also literal nail-driving)
楔を打ち込む	2821   Related literal expression (also figurative wedge-driving)

Compare daijs definitions for the two (different wording, same end-result, both making sure promises do not fall through):
釘を刺す  約束違反や言い逃れができないように念を押す。釘を打つ。「他言するなと―・される」
楔を刺す  後日問題が起こらないように、しっかりと約束を確かめる。念をおす。

book examples: drive a wedge into my brain:
僕の脳裏にくさびを刺して行くた
literal woodworking:
木材が浮いている部分があればくさびを刺して釘を打つ 
literal stoneworking:
そこにくさびを刺して、金づちで叩き込んで石を切り取った

driving a wedge between politicians:
本楔を刺して置こうとゆう構想ではあるまいか?。二、国内的にはマレンコフとフルシチフ

twitter: "drives a stake into...someone's delusion" (threatens, kills)
 そして、同時に「オタクのすべてがそうではない」と、山本氏の「オタク犯罪者予備軍説」の虚妄に楔を刺してもいます。 政治家らしい、実にバランスの

daijs absolutely gives [1].  However, searching for google book results, and twitter, gives the more common English meaning "drive a wedge into".  Can't find figurative uses outside of reference books.

Hunting for answers, I found a question about 「釘を刺す」vs 「楔を刺す」, which specifically gives an idiomatic meaning for「釘を刺す」, and a literal meaning for 「楔を刺す」.
https://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q14188525574
楔は「打つ」です。楔は敵の中に割り込んでその勢力を二分する。自分の勢力を拡大する足掛かりを他の勢力に中に作ること。
「釘を刺す」はあとで食い違いが出ないように相手に念を押すこと。

I realize this is anecdotal, and maybe "wrong", but it still reflects the obscurity of this entry.

釘を刺す on the other hand seems to mean approximately the same thing as [1], and probably a similar etymology.
https://proverb-encyclopedia.com/kugiwosasu/ 

listed as 類語 here(granted, not exactly the same, perhaps):
https://kotowaza-dictionary.jp/k1318/

This expression is [rare] no matter what glosses we give it.  But I submit that it is so rare, that it is now applied more frequently using the literal 'drive a wedge' meaning, than for whatever archaic idiomatic meaning it might have had.

"くさびを刺す" returns 2 non-dictionary book results, even fewer than with kanji.  Does not appear to be [uk]. More common on twitter as kanji as well. 0 ngrams as kana.

@@ -15,0 +16,3 @@
+<xref type="see" seq="1871830">釘を刺す</xref>
+<misc>&id;</misc>
+<misc>&rare;</misc>
@@ -16,0 +20,7 @@
+</sense>
+<sense>
+<pos>&exp;</pos>
+<pos>&v5s;</pos>
+<xref type="see" seq="2834064">楔を打ち込む</xref>
+<misc>&rare;</misc>
+<gloss g_type="lit">to drive a wedge (into, between)</gloss>

gg5, daij