User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-11227630-20170802003853/@comment-5529393-20170802042055

There's no difference. Look at the Wikipedia entry for Mahlo cardinal to see a proof that "a cardinal κ such that κ is limit and the set of regular cardinals below κ is stationary in κ" is regular.

So one can define a weakly Mahlo cardinal as "a cardinal κ such that κ is limit/regular/weakly inaccessible and the set of regular/weakly inaccessible cardinsl below κ is stationary in κ", where one can take any of the options, and the result is the same.

For you second question, adding the first "uncountable" doesn't change anything since countable ordinals are already not weakly Mahlo. adding the second "uncountable" doesn't change anything either, since all it does is remove those initial three ordinals, and stationarity is a limit property, meaning it doesn't depend on any initial part of the set. (To see this, observe that being unbounded is a limit property, since we can change any initial part of the set and it won't change whether the set is bounded or unbounded in κ.  So if a set S being stationary depended on the part of the set below a for some a < κ, that would mean there would be some club set C that intersected S, but only below a.  But then we could take the part of C that is above a; this is still a club set, but it would not intersect with S, contradicting S being stationary.)