I don't think だということ is colloquial. In fact, it's actually quite commonly used in formal or official documents. The point the OP needs to learn, and I believe it's the key he wanted to know, is that を is only attached to nouns, and copula can't be directly connected to nominalizers. That's exactly why である is used there instead of だ, or why another way of nominalizing だということ is used, as you wrote. The quotation particle also can work with 示す, and any nominalizer is not needed in this case (e.g. イギリス人だ/であると示す).
This is off-topic, and it might be better to write in your web site directly, but you posted the contents of your site in this forum and mentioned "rough" in this thread, so I want to point out that the similar "roughness" is also found in your site. You said;
When you read a present tense of verbs with Kanji.
you read its Kanji in Kunyomi, and when you read a
idiom made of 2 words(both of them are Kanji), you read it in Onyomi.
in the "What are Onyomi and Kunyomi?" section. Besides other tenses such like the past or present progressive of verbs, you'd better mention some kind of -suru verbs, for instance 信じる, 通ずる or 要する. These are on'yomi, not kun'yomi. Also, there are so many two character compound words of on-kun, kun-on or kun-kun reading such like 重箱, 額縁, 湯桶, 手帳, 南風, 川岸, etc., etc.. Your explanations are too "rough", or more likely "incorrect" or "wrong."
Furthermore, you should unify the reading of the particle は. While you wrote "konn ni chi wa" for 今日は, 今晩は is "konn ban ha" in the next line. Learners would be confused, or even could misunderstand. The same goes to "～ha nanji ni shuppatsu simasuka", "yoyaku ha hitsuyou desuka" or many other sentences. Romaji is used for those who can't read hiragana, katakana or kanji, so "wa" is more appropriate to show the correct pronunciation. My two cents.
