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I like rogue's suggestion to look at the two sentences separately.
Ok, so subjects are nouns and verbs are, well, verbs, right? And in order to have an independent clause, we have to have a subject and verb that match, and that subject and verb can't be part of some modifier, such as a prepositional phrase, a noun modifier, whatever.
The first sentence of Case 1 starts with a noun (Coach Dietz) and follows with a verb (exemplified). There's no modifying word (preposition, etc) before... so we've got a standard subject-verb setup here and, therefore, an independent clause.
The second sentence of Case 1 starts with an -ing word. -ing words are tricky - they can be nouns, but they can also be verbs, adjectives, or adverbs. If it's a noun and has a matching verb, then we could have a standard subject-verb setup. So look for verbs in the sentence. "Needed" is a verb. But that goes with the noun "we." There aren't any other verbs. So "leaving" can't be serving as the subject of the sentence because there's no matching verb to go with it.
Now let's go examine the one verb we did notice: needed. That one does have a matching noun: we. BUT we also have to look before these two words to see whether there are any "clue" words that indicate a modifier. In this case, we have the "clue" word "when." That indicates a modifier, therefore a dependent clause.
We have no other verbs with which to match a subject in this sentence... so it is dependent, not independent.
Let's take that "when" example from before and look at it slightly differently:
When she fell asleep, she snored loudly.
We've got the noun+verb "she fell" and the noun+verb "she snored." Are these dependent or independent? "She fell" begins with "when," so that's dependent. "She snored" doesn't have any "modifier clue" before it (and after the comma), so that's independent.
If you're struggling with what words qualify as "modifier clues," you might want to start keeping a list as you find them in problems. Prepositions are always a modifier clue. Who, which, where, and when are also always modifier clues. -ing words can go either way, so you have to examine those. And there are a lot of others - start actively looking for the clues and keeping lists so you can memorize them.
By the way, here's an example of an -ing word that does function as a subject in an independent clause: Riding ponies is fun. Riding = subject, is = verb. (Note: this is an unusual sentence structure, but it is valid. The speaker is not trying to say that ponies are fun; rather, the speaker is trying to say that the act of riding is fun.)
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