As a frum Jew living in Israel, I do both. My son’s talmiday Torah (that’s what they call a cheder here) has special classes and education on Yom hashoah, which is great because they are closed on Tisha BeAv. We say an extra kapitel tehillim on Yom Hashoah, and have an extra shiur or two. And we mourn them in context on Tisha BeAv. You can be a frum Jew and not be poresh min hatsibur.
At work, there is a moment of silence. Some people quietly say tehillim. But it’s a beautiful thing to see everyone stop their day and just remember, cry, feel the loss.
May we be zocheh to the geulah shelayma very soon.