The Silver collapse forcing $S(\omega, <\kappa)$ has the required properties. The conditions are functions $f$ such that $dom(f)=n\times X,$ for some $n<\omega$ and $X\in [\kappa]^{\omega}$ and $f(i, \alpha) <\alpha$ for all $i<n, \alpha\in X.$

Another example is the "universal collapse forcing" of Kunen.

For more details see Cummings paper "Iterated forcing and elementary embeddings", section 20.

Let me state (without proof) a result which shows that the Levy collapse and the Silver collapse are different.

Theorem. Let $G$ be a generic filter for $Col(\omega, <\kappa)$ and $H$ be a generic filter for $S(\omega, <\kappa).$ Then:

A. There exists $A \in [\kappa]^{\kappa} \cap V[G]$ of size $\kappa$ (which is of course $\omega_1$ of $V[G]$) such that $A$ does not contain any countable ground model set.

B. For any set $A \in [\kappa]^{\kappa} \cap V[H]$ of size $\kappa$ (which is of course $\omega_1$ of $V[H]$) there exists a countable set $B\in V$ such that $B \subseteq A$.

C. For any $A \in [\kappa]^{\kappa} \cap V[G]$ there exists $X \in [\kappa]^{\kappa} \cap V$ such that for all $Y \in [X]^{\omega} \cap V, Y\cap A \neq \emptyset.$

D. There exists $A \in [\kappa]^{\kappa} \cap V[H]$ such that for all $X \in [\kappa]^{\kappa} \cap V$ there exists $Y \in [X]^{\omega} \cap V, Y\cap A = \emptyset.$

As a corollary:

Corollary. (1). There are no Levy generic filters over $V$ in any generic extension via the Silver collapse over $V$.

(2). There are no Silver generic filters over $V$ in any generic extension via the Levy collapse over $V$.

The Silver collapse forcing $S(\omega, <\kappa)$ has the required properties. The conditions are functions $f$ such that $dom(f)=n\times X,$ for some $n<\omega$ and $X\in [\kappa]^{\omega}$ and $f(i, \alpha) <\alpha$ for all $i<n, \alpha\in X.$

For more details see Cummings paper "Iterated forcing and elementary embeddings", section 20.

Added remarks:

Theroem. (1). There are no Levy generic filters over $V$ in any generic extension via the Silver collapse over $V$.

(2). There are no Silver generic filters over $V$ in any generic extension via the Levy collapse over $V$.

So in particular the Levy collapse and the Silver collapse are not forcing equivalent. The theorem follows from the following lemma, that I state without proof.

Lemma. Let $G$ be a generic filter for $Col(\omega, <\kappa)$ and $H$ be a generic filter for $S(\omega, <\kappa).$ Then:

A. There exists $A \in [\kappa]^{\kappa} \cap V[G]$ of size $\kappa$ (which is of course $\omega_1$ of $V[G]$) such that $A$ does not contain any countable ground model set.

B. For any set $A \in [\kappa]^{\kappa} \cap V[H]$ of size $\kappa$ (which is of course $\omega_1$ of $V[H]$) there exists a countable set $B\in V$ such that $B \subseteq A$.

C. For any $A \in [\kappa]^{\kappa} \cap V[G]$ there exists $X \in [\kappa]^{\kappa} \cap V$ such that for all $Y \in [X]^{\omega} \cap V, Y\cap A \neq \emptyset.$

D. There exists $A \in [\kappa]^{\kappa} \cap V[H]$ such that for all $X \in [\kappa]^{\kappa} \cap V$ there exists $Y \in [X]^{\omega} \cap V, Y\cap A = \emptyset.$

The Silver collapse forcing $S(\omega, <\kappa)$ has the required properties. The conditions are functions $f$ such that $dom(f)=n\times X,$ for some $n<\omega$ and $X\in [\kappa]^{\omega}$ and $f(i, \alpha) <\alpha$ for all $i<n, \alpha\in X.$

Another example is the "universal collapse forcing" of Kunen.

For more details see Cummings paper "Iterated forcing and elementary embeddings", section 20.

The Silver collapse forcing $S(\omega, <\kappa)$ has the required properties. The conditions are functions $f$ such that $dom(f)=n\times X,$ for some $n<\omega$ and $X\in [\kappa]^{\omega}$ and $f(i, \alpha) <\alpha$ for all $i<n, \alpha\in X.$

Another example is the "universal collapse forcing" of Kunen.

For more details see Cummings paper "Iterated forcing and elementary embeddings", section 20.

Let me state (without proof) a result which shows that the Levy collapse and the Silver collapse are different.

Theorem. Let $G$ be a generic filter for $Col(\omega, <\kappa)$ and $H$ be a generic filter for $S(\omega, <\kappa).$ Then:

A. There exists $A \in [\kappa]^{\kappa} \cap V[G]$ of size $\kappa$ (which is of course $\omega_1$ of $V[G]$) such that $A$ does not contain any countable ground model set.

B. For any set $A \in [\kappa]^{\kappa} \cap V[H]$ of size $\kappa$ (which is of course $\omega_1$ of $V[H]$) there exists a countable set $B\in V$ such that $B \subseteq A$.

C. For any $A \in [\kappa]^{\kappa} \cap V[G]$ there exists $X \in [\kappa]^{\kappa} \cap V$ such that for all $Y \in [X]^{\omega} \cap V, Y\cap A \neq \emptyset.$

D. There exists $A \in [\kappa]^{\kappa} \cap V[H]$ such that for all $X \in [\kappa]^{\kappa} \cap V$ there exists $Y \in [X]^{\omega} \cap V, Y\cap A = \emptyset.$

As a corollary:

Corollary. (1). There are no Levy generic filters over $V$ in any generic extension via the Silver collapse over $V$.

(2). There are no Silver generic filters over $V$ in any generic extension via the Levy collapse over $V$.