Term for ability to "keep cards close to your chest"
Solution 1:
Discreet fits the bill.
Lexico
adjective
1 Careful and prudent in one's speech or actions, especially in order to keep something confidential or to avoid embarrassment.
If a quality of character is needed, discretion would be the word.
Lexico
noun
1The quality of behaving or speaking in such a way as to avoid causing offence or revealing confidential information.
She knew she could rely on his discretion