Generally speaking, you can enter dates any way you want and get the
resulting date displayed in the way how you format the cells. Formatting
the cells does not affect the input method.
"Normal" spreadsheet programs (not LO Calc) accept date input like this:
5/ --> 5th of current month
5/6 --> this year's 5th of June or 6th of May with US English locale
5/6/14 --> 5th of June 2014 according to the global settings for 2-digit
years. You can enter digits together with localized month names
according to the current locale setting. It recognizes short and long
month names of 180 languages.
In any case you should _never_ enter any weekday names.
Entering dates into LibreOffice is particularly difficult since version
3.6. To make above mentioned input patterns working in LibreOffice, you
need to adjust the "date recognition patterns" in the language settings.
The output with or without weekday names, month names 2 or 4 digit years
looks like you formatted the cells. The formatting of a cell never makes
any difference. The value remains the same.
PLEASE turn on menu:View>HighlightValues [Ctrl+F8]. This highlights all
numeric values in blue font. If your dates are displayed in black font,
they are no dates at all.
If you do not set the alignment of the cell, all numbers are right
aligned to the cell border whereas all text values are left aligned.
For any valid date in A1, =ISNUMBER(A1) should return TRUE.