So, I'm trying to solidify this idea in my head by contrasting it with the imperfecto de subjuntivo. Is the subordinate clause not in the subjunctive here because the speaker (presumably the 3rd party and the person repeating the statement) take for granted the factual of the idea (ie in the sentence "el hombre de tiempo dijo que llovería hoy" that the idea that it is going to rain is considered a fact, and not a supposition.
I'm not sure if you are trying to find a difference between the two different meanings of "decir que". You can use "dice que, "dijo que" followed by the present subjunctive or the Imperfect subjunctive but with a different purpose, that being an "order/request", the same way you can say in English for example "he told/asked me to go away": Me dijo que me fuera.
This usage of decir is different to the one in the lesson. The one mentioned above requires the subjunctive because of the functionality of the verb decir here (order, request).
In this lesson using decir que plus the conditional, the conditional could be stating the obvious/a fact/something that is actually happening in the future, like the example about the weather forecast, but even if that is not seen as a fact or something that may or may not happen, we could still use the conditional:
El hombre del tiempo dijo que hoy llovería.
El hombre del tiempo dijo que quizás hoy llovería.
This is more to do with the indirect speech and the sequence of tenses:
The forecast man says today:
Tomorrow it will rain
Then the following day you report what he said:
The forecast man said yesterday that it would rain today.
El hombre del tiempo dijo ayer que hoy llovería.
When we report about something that is using the future tense in the direct speech, we use the conditional.
I hope this clarified it.
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