I think most people would generally agree, and it's hard to say whether they actually want to depend on this level of nuance, but "speak Korean" and "speak in Korean" are slightly different.
Adding a subject helps to illustrate the difference: "We speak Korean" serves less to describe an action, and more to describe an ability of the subjects. Whereas "We speak in Korean" is more of a momentary thing, that describes particular actions.
That said, I also agree that most people don't care about this level of nuance, so they should probably just accept both
어떨 때 means “in what kind of situation”, “in what kind of times” or even just “when”. However, 어떨 때 can also be used to mean “sometimes”. But this is when you want to show contrast between “some times” and “other times”. Therefore, you usually use 어떨 때 with the topic marker, -는.
(1) typo. 대 = stage; platform etc.; 때 = time; occasion
(2) My guess:
(i) #어떤 때 -> 어떤 has 2 meanings:
• 어떤, as general determiner, some; any
• 어떤, as (present tense) attributive adjective deriving from the verb 어떻다: being how/being in the state
=> 어떤 때 =
• at one time/ a certain time;
• on some occasions; some of the time.
The topic marker 는 is used to introduce "contrast".
어떤 때는 means some of the time, but not always
어떤 때는 우리가 한국어로 말해요 = Some of the time, we speak in Korean [i.e. on some occasions, we do and other occasions, not.]
Note: 어떤 때는 Some of the time and 가끔 sometimes are virtually synonymous. The nuantial difference is 어떤 때는 carries a contrasting undertone.
(2) #어떨 때
어떨 like 어떤 is a determiner; / attribute deriving from 어떻다 but in the future tense, that is to happen/will happen. It implies certain expectation from the speaker.
어떨 때 = when/on which occasion(s) that will happen i.e. when things are expected to happen = at some vague point in time; sometime; when.
With 는 (contrast), 어떨 때는 takes on the meaning of usually (but not now) (?)
• 어떨 때는 우리가 한국어로 말해요 = We usually speak (but not now) in Korean
Note: I am not 100% comfortable with the interpretation of 어떨 때는 as usually. I tend to understand it as some/ another time maybe (but not now).
Perhaps, someone can help?
It was in the website TTMIK, link he provided... . https://talktomeinkorean.libsyn.com/webpage/2011/10/17