Is there a difference between a Shimano 10-speed MTB small chainring between double and triple cranksets?
I have a list of 9-12 speed Shimano chainsets here:
https://docs.google.com/spreadsheets/d/1HdcNgmQKFA-lAqQJ3lXQ8SwA4tblAKiOXe9XYJJHYN8/edit#gid=375920416
Refer to the 10-speed MTB tab.
There is considerable overlap between double and triple small chainrings for MTB (none for road).
E.g. SLX FC-M677 is either 38/24 or 36/22, meanwhile SLX FC-M672 is 40/30/22.
Recall that the large ring has aids to pick up from the smaller

but the small ring needs no assistance to have the chain fall off the larger.
So the larger ring matches with a specific smaller
The 10-speed 40-30-22 triple is referred to by Shimano as 'AN', hence a 30t AN means a 30t 10-speed picking up from a 22t. https://si.shimano.com/api/publish/storage/pdf/en/ev/FC-M672/EV-FC-M672-3588.pdf
Since it is confusing to buy the right large/middle ring, Shimano over time would tend to differentiate things that perhaps are not different.
If we look at the M677 double, then it is
https://si.shimano.com/api/publish/storage/pdf/en/ev/FC-M677/EV-FC-M677-3911.pdf

Y1KF22000 Chainring 22T-AA/AC/AY
Y1NA24000 Chainring 24T-AM/AZ
Y1RM98010 Chainring 36T-AY
Y1RM98020 Chainring 38T-AZ

I.e. the 36t is AY picking up from 22t, which could in fact be 'AA', 'AC' or 'AY' (the same identical ring), and likewise 38t AZ is picking up from 24t which could be AM or AZ
Looking at the 22t part number
https://bike.shimano.com/en-AU/product/service-upgradeparts/slx-m670/Y1KF22000.html
it turns out that the identical chainring is used not only on double 10-speed chainsets but also on both double and triple 9-speed chainsets
The 'AA' pattern refers to a 9-speed 44/32/22 triple https://si.shimano.com/api/publish/storage/pdf/en/ev/FC-M970/EV-FC-M970-2592.pdf
And 'AC' appears to be an alternate (to AY) name for a 36/22 double
https://si.shimano.com/pdfs/ev/EV-FC-M3000-B2-4286.pdf
So the answer is that the inner chainrings are the same, and very tolerant of differences, while middle/outer rings are less so. The 2x and 3x checkboxes should be to help with identification - since there are three envelopes:

22t-30t

32t-42t


44t-53t



In this context, '36t' is clearly highly ambiguous, so by saying '2x' or '3x', we can distinguish between a 36t middle ring and a 36t outer ring. This turns out to still be ambiguous in some cases, as there are 36t rings from 24t and 36t from 22t, but it at least helps. The exact part number or two letter code is the way to fully resolve whether it is the correct chainring
However for inner rings (and for 1x drivetrains), these are provably the same part number across a wide range of cranksets, including, in some cases, the same inner chainring used for triple and double cranksets.