Indefinite articles usage http://www.oxfordlearnersdictionaries.com/us/definition/english/a_4 go down to #10: used before the names of days of the week to talk about one particular day
She died on a Tuesday.
Yes compare the meaning of "she died on Tuesday" - someone died literally last Tuesday "she died on a Tuesday" - any Tuesday in the past, probably a long time ago
"they're getting married on Sunday" - the wedding is literally Sunday this week "they're getting married on a Sunday" - their wedding is on some future Sunday
The sentences with "a" are highlighting the day something happened/will happen. For example as if they thought it was weird or notable to get married on a Sunday.
Yes, it is a mistake. Dropping the "a/ein" changes the meaning.
However, it would be normal to drop the "the/der" (definite article) which is used in German with dates, e.g.:
Ich werde an dem Samstag da gehen = "I will go there on Saturday"
Der Juli ist ein guter Monat dafür. = "July is a good month for that."
zengator: In this case the prompt included 'ein', which sounds perfectly normal to me, like it would in English if referring to a specific Saturday. But like in English, where depending on the context you can drop the 'a', can one say in German 'Das ist Samstag' or does that not mean 'This is Saturday'?