Well as in the online Shuruh (of as-Sindi or as-Suyuti) of Sunan an-Nasa-i apparently no scholar even thought of explaining the meaning of the word you refer to as "to play" based on the translation provided by sunnah.com, which may mean that people would understand it or that those scholars didn't take much care of the matn rather then on the sanad (maybe because an explanation of that comes along in a hadith mentioned earlier or later). And as my own database is unfortunately in a HDD which seems to be destroyed.
I made a small investigation on the meaning of **لعب**in Arabic. But before here's the Arabic original from this hadith:
كَانَ لَكُمْ يَوْمَانِ تَلْعَبُونَ فِيهِمَا وَقَدْ أَبْدَلَكُمُ اللَّهُ بِهِمَا خَيْرًا مِنْهُمَا يَوْمَ الْفِطْرِ
And from the given meanings of almaany we may conclude:
That maybe the translation "to play" is a bad choice as what scholars may talk about or define as a halal way to pass a 'Id is anything which we may call (halal) amusement so the meaning is more likely to be:
- fun or
- diversion or
(in a halal manner).
This could also be concluded from this two ahadith in Sunan Abi Dawod and Sahih al-Bukhari.
This Link provides a more clearer explanation for what "to play" may refer to:
المقصود باللعب هاهنا: ألوان المتعة واللهو، مثل اللعب بالسهام، والقسي
والرمي، والنبل، والسيوف، والاستعراضات، فيبدو أن لعبهم بالنسبة للرجال
كان أشبه بالاستعراضات العسكرية، مثلما كان الحبشة يلعبون في المسجد
ويزفنون، وقد يكون فيه نوع من الرقصات الرجالية.وبالنسبة للنساء كان
أيضاً لهن حظهن من اللعب مثلما في حديث عائشة،
So for men: it includes playing with arrows and weapons maybe like we may understand a military review nowadays, playing instruments and dancing (even in the mosque), which goes ahead with the hadith reported by al-Bukahri and Muslim in their Sahihs! And the women also had a part of that amusement as we can read for example in Sahih Muslim and al-Bukhari.
And Allah knows best!