写 (traditional 寫) is 宀/冖 (a roof or cover) over a phonetic part. The old sense was 'to place something under cover / to set down' — which became 'to copy out, to write'. The simplified form keeps the little cover on top.
写 is the active half of literacy: 写字 (to write characters), 写信 (write a letter), 写书 (write a book). Chinese handwriting is its own art — 书法 (calligraphy) — where HOW you write a character matters as much as which one. Every stroke you trace in Zhizhi is you learning to 写, not just to read.
Setting something down under a cover (冖) — to WRITE it down.
写字 is to write; 书法 (shūfǎ) is the art of writing beautifully — calligraphy, prized for millennia. The stroke-order practice in Zhizhi is the first step toward both.