How to understand the meaning of "in which" and "on which" when seeing them standing at the beginning of questions?

There are two essential considerations to take into account and these are the preposition used habitually with the verb that follows and the preposition habitually used with a noun that follows, whichever applies.

In the sentence "In which two ways are the lines distinguished from each other?" a following noun is "way" and a following verb is "to distinguish" (in the passsive voice). One way to make out the proper preposition is to build the answer to the question. To do so you'd start like this.

The two ways are X and Y.
Having done so you see that you can make a more complete answer and mention what those ways are about. To do so you have to introduce that
1 they are the ways to distinguish the lines from each other or, put another way, that
2 they are the ways of distinguishing the lines from one each other. or put yet in another way, that
3 they are the ways in which the lines are distinguished from one another

This tells you what the right preposition is. If you formulate a "what" or a "which" question you have three choices.

• What/which are the two ways to distinguish the lines from each other?
• What/which are the two ways of distinguishing the lines from each other?
• In what/which two ways are the lines distinguished from each other?

Of course, you must know that when using the passive voice the preposition is "in", which requires a certain familiarity with the language.
You can see that in the first two cases the preposition is the usual preposition for introducing a prepositional complement of the noun "way", whereas in the last one it is the usual preposition used to build an adjunct with the noun "way".

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In the sentence "On which type of transport can a one-day travel card be used?" the answer is "A one-day travel card can be used on/in several types of transport; those are…"

So the preposition is "on", but it could be "in" too.