"Silakan" (e.g. Silakan minum air / masuk) と "Tolong" (e.g. Tolong minum air / masuk) と "Mohon" (e.g. Mohon minum air / masuk) はどう違いますか？説明が難しい場合は、例文を教えて下さい。

@biotachyonics Could you also take my question written in the additional form? I want to reconfirm my interpretation in these case studies is correct.

@Gwi93 Could you also take my question written in the additional form? I want to reconfirm my interpretation in these case studies is correct.

Thank you all :) Everybody here is sooooo informative. Below is the summary of my misunderstanding:

1) "Tolong minum air" in the drug prescription advice. The verb is better said "diminum" (be drunk) than "minum" (drink).

2) "Tolong masuk" or "Mohon masuk" in the doctor case. To enter in the clinic (or the doctor's examination room) is totally up to patients' will. So, "silakan" is the right choice. "Tolong masuk" can be used for a conversation between a teacher and a student as tacok explained. I guess the same pattern can be applied between a master and her housekeeper. -- I (Ms. Fixer) am working in a study room. My housekeeper knocks the door to ask me whether she can clean the room or it disturbs me. And then, I would say "Tolong masuk" to give her the green-light (Now is okay. Please come on in and clean the room).

It's a bit difficult to come up with a good sample sentence with "Mohon masuk"... Maybe "Mohon jangan pergi dariku!" (Please don't go away from me! I still love you...) would be better for lyrics or poems (i.e. written format with "mohon" instead of "tolong").

I think I understand now how to use these three words in which situations. Thank you!!

In return, I would like to point out that many of you here totally misunderstand the translation from Indonesian to Japanese...

「どうぞ」is not always "silakan".
「ください」 is not always "tolong".
「お願い致します」「お願い」 is not always "mohon".

Bahasa Indonesia distinguishes "silakan/tolong/mohon" based on eagerness (i.e. how much the speaker wants the listener to execute the action: offering << asking << begging).
However, the abovementioned Japanese phrases are used based on TO WHOM the speaker is talking basically. These Japanese words are not different in terms of eagerness of the speaker. Let me recycle the case study in order to explain my point.

Unlike "mohon", 「お願い致します」is used both in writing and speaking. It means "asking" only, and we don't say 「お願い致します」when we beg. 「お願い致します」is used when the listener is customers/clients/suppliers in business (i.e. stakeholders), or unfamiliar people in general no matter of ranking/social status/age.「お願い致します」is an indicator of a sort of "social distancing" in a polite way.

Moreover, 「お願い致します」sometimes even means "with regards," or "sincerely" as the last sentence of a letter used by English speakers. So,「お願い致します」sometimes functions as a polite greeting when you walk away from the conversation/messaging. It's not so serious as you imagine. Thus, connecting「お願い致します」with "mohon" doesn't make sense.

「お願いです」can be used as "mohon" (begging situations) or "tolong" but only when 1) the listener's status/position is higher than the speaker; or 2) it's a public announcement. For example, I (Ms. Fixer) was kidnapped and in captivity by a bad guy. I want to be released, so I beg the bad guy, saying「お願いです。解放してください」(I'm begging you. Please let me go). I have no liberty and the bad guy is controlling the situation. That's why I say「お願いです」. -- As you can see,「ください」and「お願い」are used in the same context. Meanwhile Indonesian people don't use "tolong" and "mohon" at the same time because you don't ask and beg at the same time.

Another example for a public announcement:「当店からお客様にお願いです (or お願い申し上げます)。トイレはきれいにご利用ください」-- This announcement means "This is a request from our shop to our valuable customers. Please keep our restrooms clean when you use (e.g. don't scrabble or don't get clogged etc.)". 「お願い」and「ください」are used either on a notice board or over the PA (i.e. writing and speaking). Obviously, the shop is not begging its customers. So you cannot translate「お願い」as "mohon".

The closest connotation of「どうぞ」is "please go ahead", "as you wish" and "here you are". The phrase "Please come in" is the best case to explain the difference among Bahasa Indonesia, English and Japanese.
You say "Silakan masuk" only for the home welcoming or doctor-patient cases. But we Japanese say「どうぞ」(we usually omit お入りください as in enter/come) for the home welcoming, the doctor-patient, and the master-housekeeper cases. Apparently「どうぞ」is NOT used based on eagerness. In the master-housekeeper case, my housekeeper knocks the door to ask whether it would disturb me. My answer as her master is "Disturbing? No, not at all. Please come in as you wish", so I say「どうぞ」in Japanese. Of course I am not welcoming my housekeeper as a guest. So, I can't say "Silakan masuk". Therefore, connecting「どうぞ」and "silakan" doesn't make sense.

You may now understand why I wrote such a set of long messages. Indonesian, Japanese and English speakers have totally different cultures, and understanding the gap is very interesting to me.

@biotachyonics Thank you for additional sentences. I'm sorry but I'm overwhelmed... Do you disagree with the other native responders' explanations? Below is the summary of their points:

1) "Tolong" (as in "please") is used more frequently than "mohon". "Mohon" is seen in writing mostly.
2) If you rephrase from "tolong" to "mohon", the whole sentence increases the degree of eagerness (i.e. from "asking" to "begging").
3) "Mohon" is often followed by "jangan" (i.e. negative imperative sentence).

Some of your additional sample sentences suggest that "mohon" is also used for "asking". But you didn't say it is in writing or speaking.

I would like to know the difference of nuance and in which situations I can and cannot use these phrases. That's why I set "minum air" and "masuk" cases in the original question.

@de_sus Hahaha, I'm not a researcher, linguist or teacher. I just want to be polite when I ask someone to do. I have been to Indonesia several times, and I found that Indonesian people in general are friendly and full of hospitality. They are willing to help foreigners like me. But they also get intimidated easily when they are scolded or ordered in front of others. Using wrong words to ask for help may harm their self-esteem.

It took a while to decide which answer should be featured... I usually search for others' Q/A here on HiNative before posting my own question. So, this time I chose the answer taking my first question clearly in order for other fellow learners to make their search short. @Agatha_20 's answer is relatively short, but makes a clear point. The featured answer thereby goes to her answer.

I also thank all of you for giving different points of view and sample sentences. I am going to blog about this experience to summarize and share with others, especially with Japanese speakers who do not understand English well.