steinern
made of stone
Made of or consisting of the material stone.
Eine alte, steinerne Brücke führte über den Fluss.
An old stone bridge led across the river.

stony
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made of stone
Made of or consisting of the material stone.
Eine alte, steinerne Brücke führte über den Fluss.
An old stone bridge led across the river.

The adjective 'steinern' (stony/made of stone) is formed from the noun 'der Stein' (the stone) and the suffix '-ern'. This ending often means 'made of the material ...'.
Der Wanderweg führte über eine steinerne Treppe den Berg hinauf.
The hiking trail led up the mountain over a stone staircase.
The meaning of 'steinern' (stony/made of stone) depends on the noun. If it describes an object like a wall or bridge, it means 'made of stone'. If it describes a facial expression, it means 'emotionless'.
Er hörte die schlechte Nachricht mit steinernem Gesicht.
He listened to the bad news with a stony face.
In the figurative meaning, 'steinern' (stony) is very often used in the fixed combination 'eine steinerne Miene' (a stony expression). This describes a facial expression that shows no feeling at all.
Während der gesamten Verhandlung behielt der Angeklagte eine steinerne Miene.
Throughout the entire trial, the defendant kept a stony expression.
cold
'kalt' and 'steinern' can both describe someone showing no feelings. 'Kalt' is more about the effect on others: a person seems distant, unfriendly, or lacking compassion. 'Steinern' especially describes an unmoving expression, often on the face. It is less about whether someone is unfriendly and more about the fact that no reaction can be seen. So a person can speak coldly but have a stony expression.

Vor dem Amt wartete Tom mit steinerner Miene auf die Entscheidung.
In front of the government office, Tom waited for the decision with a stony expression.
Der Chef antwortete kalt, obwohl die Mitarbeiterin sehr nervös war.
The boss answered coldly, even though the employee was very nervous.