GG5: 	medals for merit.
中辞典: 	(the class of) an order.
ルミナス: the order of merit.

The JEs are a bit all over the place. I think the sense split works. Trimming a bit.

@@ -13 +13 @@
-<gloss>an order of merit</gloss>
+<gloss>(an) order of merit</gloss>
@@ -18 +17,0 @@
-<xref type="see" seq="1247200">勲一等</xref>
@@ -20,2 +19,2 @@
-<gloss>order (of medal, decoration, etc. i.e. first order, second order, etc.)</gloss>
-<gloss>class number (first class, second class, etc.)</gloss>
+<gloss>order (first, second, etc. of a medal or decoration)</gloss>
+<gloss>class number (first, second, etc.)</gloss>

https://kotobank.jp/jeword/勲等
progressive: "an order of merit

Wikipedia identifies "勲等" as the class number here:
In the system before the 2003 revision, titles consisted of two parts, the class number (kuntō [ja]) and the decoration. For example:
勲一等旭日桐花大綬章 (literally: 1st Class, Grand Cordon of the Order of the Rising Sun, Paulownia Flowers)[16]
勲一等旭日大綬章 (literally: 1st Class, Grand Cordon of the Order of the Rising Sun)[16]
勲二等旭日重光章 (literally: 2nd Class, The Order of the Rising Sun, Gold and Silver Star)[16]



Meritocracy and Its Discontents:
Long-run Effects of Repeated School Admission Reforms
Mari Tanaka Yusuke Narita Chiaki Moriguchi∗
November 30, 2021
https://ies.keio.ac.jp/upload/20221014appliedpaper.pdf
In the pre-WWII Japanese honor system, the the medals for merit (kuntou in Japanese)
and the court ranks (ikai in Japanese) were conferred on individuals in recognition of their
exceptional public service or distinguished merit. The medals consisted of 8 grades from the
first order of merit (the highest honor) to the eighth order of merit (the lowest honor), and
the court ranks consisted of 16 ranks from senior first rank (the highest rank) to junior eighth
rank (the lowest rank). According to Ogawa (2009), the highest honors are given mostly
to public servants (such as top-ranking military officers, bureaucrats, and politicians), while
individuals in a private sector (such as top corporate executives) receive the fourth order
of merit and below. Given this, we define prestigious medal recipients as those who receive
at least the fifth order of merit (kungotou in Japanese) or the junior fifth rank (jugoi in
Japanese) .

I believe the numbered system identified by [1] has essentially been replaced. If you look here, you'll see an alignment between 勲等 system and modern naming:
https://kotobank.jp/word/勲等

reverso has a lot of "medals for merit".  Not sure how I feel about this, but it is the translation used by a scholarly (prepublication, don't know what for) Japanese paper from Keio.

The wikipedia xref is to the whole system of Japanese honors. progressive (book result) says "an order of merit", which was my initial thought.  Big difference between "an order of merit" and "the order of merit". sankoku's definition still seems correct, which would be sense[2].  Even this usage seems to have been deprecated in recent years, with the classes/orders being completely dropped.

@@ -11,0 +12,6 @@
+<xref type="see" seq="1155420">位階勲等</xref>
+<gloss>an order of merit</gloss>
+<gloss>medals for merit</gloss>
+</sense>
+<sense>
+<pos>&n;</pos>
@@ -14 +20,2 @@
-<gloss>order (of medal, decoration, etc. i.e. first order, second order, ...)</gloss>
+<gloss>order (of medal, decoration, etc. i.e. first order, second order, etc.)</gloss>
+<gloss>class number (first class, second class, etc.)</gloss>

https://ja.wikipedia.org/wiki/勲等

Ok, I got ahead of myself. Possibly wrong, possibly both?  I will update again in a bit...
I started this edit because the wikipedia xref was (formerly anyway) "Orders, decorations, and medals of Japan".  sankoku's gloss matched what I was reading on some other honors.

勲等	3586	16.9%
勲一等	10213	48.1%
勲二等	3401	16.0%
勲三等	4036	19.0%

勲一等、　勲二等、　勲三等、first, second, third order...

This isn't "The Order of Merit", it is "the order of the merit". 

sankoku: 勲章の等級

@@ -12 +12,3 @@
-<gloss>the Order of Merit</gloss>
+<xref type="see" seq="1247200">勲一等</xref>
+<xref type="see" seq="1247240">勲章</xref>
+<gloss>order (of medal, decoration, etc. i.e. first order, second order, ...)</gloss>