Off-ball/on-ball vs off-the-ball/on-the-ball. Which versions are correct as adje...

In football (soccer), the expression "on the ball" or "off the ball" is often used to indicate whether a player is in possession or not in possession of the ball. It is also often used as an adjective, e.g.,

off-the-ball movement, on-the-ball event.

However, I have encountered that sometimes shorter "off-ball" or "on-ball" expressions are used as adjectives. Therefore, I wonder if there are some strict rules behind the presence (or absence) of "the" within the compound adjective.

• 1. Is it only brevity that favours shorter versions sometimes?
• 2. Is one more popular than the other?
• 3. Does it depend on the formality of the text? Especially, should one be favoured over another for academic writing?

Therefore, I wonder if there are some strict rules behind the presence (or absence) of "the" within the compound adjective.

There are no "rules" in English: there is guidance.

1. Is it only brevity that favours shorter versions sometimes?

Yes, that and the choice of the speaker.

2. Is one more popular over than the other?

Compare them on Google Ngrams, but popularity does not equal being "idiomatic", "natural" or "appropriate."

3. Does it depend on the formality of the text?

On/Off-the ball is slightly more formal but does not really reach the criteria for "formality."

Especially, should one be favoured over another for academic writing?

You will have to provide an example of "academic" context.