You have understood some of the subtler meanings that can be expressed in the demands that you've listed. But be careful, because those demands don't always have the meanings you've described. You still have to pay attention to the context and the tone of the expressions in order to understand the finer shades of unspoken meaning that the speaker is expressing.
'Obscure' is not quite the right description of what the word 'just' does in the last two cases to the meaning of 'at once', or 'immediately'. The word 'just' implies that this is not the first time you've made the demand. So you're right that 'just' can add the meaning, 'don't play around any longer'. However, these are not playful phrases, so they would be used very rarely with 'just'. To obscure something means to conceal or hide it.
These two are identical in their literal meaning:
Tell me at once.
Tell me immediately.
and either one can express both of the nuances you've ascribed to them. If used by themselves the listener will almost certainly feel talked down to, as if you considered them inferior or childish. Instead, I would suggest
I need to know at once.
I need to know immediately.
Please tell me at once.
Please tell me immediately.
I don't know how long you've been speaking English. I think you're making good progress in understanding subtle, unspoken meanings. But keep observing because you will discover ever more subtle knowledge about the use of those expressions the more you listen.
Also, be careful when using demands like these. As Weather Vane has commented, people will generally not appreciate being spoken to with that kind of language. You're much better off using the 'nuances' themselves, or adding polite phrases like, 'Could you please...'. Your tone of voice will make a big difference in how you're understood. A demand can usually be made playfully without negative consequences, but when speaking seriously or with someone you don't know well it can easily be taken negatively and met with hostility.