When you standard paste (cntl-v) onto a cell like that, you are pasting everything, including the format.

When you copy "2/4" from here, you are copying the text and the format though the format is a null (for lack of a better term). So when you standard paste, Excel enters the data and sees the null format. It then assumes date and formats it as such. It can be quite frustrating as you know. However, it is a time saver most of the time.

You can get around this with a special paste:
* Make sure the cells are preformatted as text.
* Copy the text.
* Click the dropdown below the Paste button. You will see two icon options. If you hover over each, they say
----Keep source formatting (M): this is the standard paste
----Match destination formatting (K): this is a text only paste
* Select the latter, and it will display the way you are looking for.

I'm not really sure what the letters in parentheses above signify. They don't appear to be shortcuts.