[rare] can be used for a single sense but in most cases another tag is more appropriate. I think this sense is either [dated] or [obs]. GG5 has it but daijs's example is from 1877.
I don't think the 退学処分 x-ref is needed. It's not interchangeable with 退学.

@@ -27 +26,0 @@
-<xref type="see" seq="1779190">退学処分</xref>

[rare] is more for the whole entry.

@@ -35,2 +35,2 @@
-<misc>&rare;</misc>
-<gloss>leaving school (university, etc.) after completing a course of study</gloss>
+<misc>&dated;</misc>
+<gloss>leaving school (university, etc.) after completing a course</gloss>

https://ja.wikipedia.org/wiki/退学

退学	280974	92.8%
退学処分	21756	7.2%

退学 alone　is predominantly [1]
https://context.reverso.net/translation/japanese-english/退学

*lots* of glosses and circumstances, none of them are [3].
「退学」は英語で何という？退学方法で変わる表現を確認しよう！
https://nativecamp.net/blog/20230510-leave_school

eijiro has two pages of collocations, none look like [3]
https://eow.alc.co.jp/search?q=退学

処分 = "punishment" nuance here,I think. it is the example in sankoku's 退学 entry. On reverso, 退学 is predominantly [1], 退学処分 is unambiguous.

sankoku does not give [3] at all, says explicitly 途中でやめ. Wikipedia also says 卒業・修了を待たずに学校を途中でやめること

[3] seems like it would be an uncommon usage, given the nuance involved in [1] and [2]. Also, dropped from modern dictionary (sankoku). It's obviously literally correct for A+B reasons, 退+学, but unlikely to be common in this day and age. (marking [rare] or [hist] or [dated] explains that 退学 is not as ambiguous as the 3 senses might otherwise indicate).

@@ -26,0 +27 @@
+<xref type="see" seq="1779190">退学処分</xref>
@@ -34 +35,2 @@
-<gloss>leaving school (university, etc.) after completing a course</gloss>
+<misc>&rare;</misc>
+<gloss>leaving school (university, etc.) after completing a course of study</gloss>

meikyo

@@ -18,0 +19 @@
+<pos>&vi;</pos>
@@ -20 +21 @@
-<gloss>withdrawing from school</gloss>
+<gloss>withdrawal from school</gloss>
@@ -24,0 +26 @@
+<pos>&vi;</pos>
@@ -30,0 +33 @@
+<pos>&vi;</pos>