I don't think the accepted answer is accurate enough. My answer covers broader aspects of these two words, too.
First notice 「に」 creates a general adverbial phrase, while 「から」 means "since" or "(starting) from", so
- 「○○ (a clause) 内に(うちに)」 means "while ○○ is still the case", and
- 「○○内から(うちから)」 means "starting from when ○○ is the case".
(For searchability, I here use both 平仮名 and 漢字 forms.)
Practically, 「内に」 is often used when the current situation is good, and it is likely to be lost. 「彼がこっちに居る内に」implies "It'd be too late if it's after he has left". Thus there can be the speaker's opinion / judgment in 「内に」.
But there's also other cases. 「熱い内に触っちゃった」 means "I mistakenly touched it while it was still too hot".
Similarly to my first example, 「うちから」 often implies 'it may seem too early'. 「晴れている内から始めておこうと思う」 = I'm going to begin it while it is still sunny (to be proactive, though it can perhaps be later.) 「業績が良いように思われていた内から、実は財務状況は最悪だった」 = "As early as that time when people thought that their performance was good, but in fact the finance was already terrible." (Notice I attached "already" to the second clause, rather than putting "still" instead to the first clause. Such phenomenon is common in translation.)
In the sentence of your question, she is angry, because Dad's assertion was indeed too early (so in fact groundless), when nothing has happened yet.
BTW your translation of 「わざわざ」 is more natural than "go out of your way" in the accepted answer.
Sometimes 「うちに」and「うちから」 can be interchanged: 「晴れている内に始めておくべきだった」 means exactly the same as 「内から」, but not always. 「うちから」 is leaned toward starting, and can never be used with termination. 「今の内に終わらせる」 can not be rephrased as "今の内「から」終わらせる".
The examples of 「おとうさん...」「業績が...」can not be used with 「うちに」, but why? It's difficult to answer, and I have to admit the following explanation is not good enough, but うちに seems to mean perfection, (rather than completion.) 「今の内に始める」 means "I will have begun it soon". Yes, it's unnatural as English, but not wrong either. It becomes clearer IMO if you shift it to the past conditional: 朝の内に始めれば良かった = I should have begun it in the morning. (Looks like past perfect.)
In addition to their conjunction uses, there's also 「noun + の + うちに / うちから」 forms. Top common uses are 今の / 今日の / 朝の + うちに / うちから。
One more difference is that 「うちから」as a conjunction sounds a little old-fashioned - Your citation has 昭和 era taste (added with its "too feminine" discourse.) Today more ordinary phrasing will be 「まだ何も起こっていないのに、わざわざ断言することないでしょ!」「業績が良いと思われていた時には既に」 OTOH 「nounの内から」is common.