Cordiality
Etymology
From Medieval Latin cordialis "of the heart," from Latin cor "heart" — the same root behind "cordial" the heart-stimulating drink. Cordiality is literally heartfelt-ness: warmth issuing from the heart rather than mere form.
noun · /kor-jee-AL-uh-tee/
Sincere, friendly warmth in how one treats others.
What it names
A handle for warmth that is genuine and heartfelt yet not necessarily intimate — the pleasant graciousness that makes an encounter feel welcome.
Example
The rival CEOs greeted each other with a cordiality that surprised the reporters expecting fireworks.
Related idea
Distinct from civility: civility is correct but can be cold, the bare minimum of respect; cordiality adds genuine warmth. Yet cordiality stops short of intimacy — you can be cordial with a new colleague or an old opponent without being close to either.