Is there a difference between these types:

 INTEGER (1..10) (MIN < .. 8)

and

 INTEGER (1..10) (MIN .. 8)

There are not equivalent. In both cases, MIN is the minimum value from the previous set, that is, 1, but 1<..8 means any integer starting from just after 1 through 8. In other words, 1<..8 is the same as 2..8.

Would 1 be a compatible value for both of these types, or just the
second type?

1 is only valid for the second case.

In other words, does the open-ness of an endpoint matter when
considering MIN and MAX?  I'd guess "yes" from X.680, but maybe someone
knows differently.

It depends upon the value MIN or MAX is taking. Something like

   INTEGER (MIN .. 8)

means the same thing as

   INTEGER (MIN <.. 8)

since MIN is minus-infinity and one higher than minus-infinity is still minus-infinity. On the other hand, for

   INTEGER (1..10) (MIN..8)

MIN would be a 1 and so

   INTEGER (1..10) (MIN<..8)

would range from just after 1 through 8 or 2..8.

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