I understand the question as to how to differenciate to express in Japanese the differences in nuances of sentences using various causative verbs in English.
Before answering your question, I visited the site here and learned how causative verbs are explained in English.
FORM: [let + person + verb]
USE: This construction means "to allow someone to do something."
FORM: [make + person + verb]
USE: This construction means "to force someone to do something."
FORM: [have + person + verb]
USE: This construction means "to give someone the responsibility to do something."
FORM: [get + person + to + verb]
USE: This construction usually means "to convince to do something" or "to trick someone into doing something."
Get vs. Have
Sometimes "get someone to do something" is interchangeable with "have someone do something," but these expressions do not mean exactly the same thing.
I got the mechanic to check my brakes.
At first the mechanic didn't think it was necessary, but I convinced him to check the brakes.
I had the mechanic check my brakes.
I asked the mechanic to check the brakes.
If you read the explanation for the four causative verbs, you can see that it is "make" and "get" that the expected results are executed, apart from the person's feelings to execute something.
Therefore, I think the questioner is smart to list these two in the example of comparison.
Get Mr Tanaka to tell you that.
Make Mr Tanaka tell you that.
As for phrase 1 having "let" in it, "田中さんに教えてもらってください" is cosiderably good as a Japanese translation, but I think it is insufficient whether the expected result will surely be obtained or not in it. In order to ensure the result and to give the nuance of "to convince to do something" or "to trick someone into doing something" that was explained in the previous English grammer textbook to it, "説得｛せっとく｝する", "納得｛なっとく｝さす", "言｛い｝い含｛ふく｝める" and "言｛い｝い包｛くる｝める" and so on seems appropiate to be added to your attempt. But all of these options are too exaggerated compared to the simple verb "get" in the original phrase. So I think "是非｛ぜひ｝" or "是非｛ぜひ｝とも" is appropriate to be added like "田中さんに是非とも教えてもらってください" to make it have a similar nuance that the original English phrase has.
As for phrase 2 having "make" in it, the interpretation in Japanese seems to have problems.
As is said in the Chocolate's answer, 教えさせてください sounds like the person taught by Mr. Tanaka will be someone other than the hearer, and more, it sounds unnatural for other reason. The reason that makes the Japanese interpretation unnatural is the use of "教える", whose main meaning is, as you know, not " to tell" but "to teach".
Because "何かを教える人 a person to teach something" is generally in a position respected by "教えられる人 a person to be taught:", 教える人 is not compatible with the nuance of "a person to be forced to do something". Therefore, in order to force Mr. Tanaka "to tell you something", and also in order to surely obtain the result from him, I would change expressively the Japanese translation for "to make someone tell" like "言わす/言わせる", "伝えさす/伝えさせる" or "聞き出す" instead of "教えさす".
As a translation to solve the two problems, one is what is pointed out in Chocolate's answer and the other is the difference in a nuance between "教える" and "tell", I propose the following interpretations like: "それは田中さんに言わさせてください" or "それは田中さんから聞き出してください".