It seems that the answer to the revised question is yes: categories parameterized over $O_G$ are the same as categories internal to $Top_G$.

Let's think through this carefully. A category object should satisfy Segal conditions and a univalence (aka completeness in the sense of Rezk) condition. Admittedly the latter might not be entirely standardized for internal categories, but let's pick a formulation and run with it:

• A category parameterized over $O_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ which is, levelwise in $O_G$, a complete Segal space. So we have

1. Segal conditions: $X(G/H,[n]) \to X(G/H,[1]) \times_{X(G/H,[0])} \dots \times_{X(G/H,[0])} X(G/H,[1])$ is an equivalence.

2. Univalence: $X(G/H,[0]) \to Map(E_\bullet,X(G/H,\bullet))$ is an equivalence of spaces.

Here $E_\bullet$ is the (nerve of the) walking isomorphism, and $Map$ the $Top$-enriched homset of simplicial spaces.

• A category internal to $Top_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ satisfying

1. Segal conditions: $X(-,[n]) \to X(-,[1]) \times_{X(G/H,[0])} \dots \times_{X(-,[0])} X(-,[1])$ is an equivalence.

2. Univalence: $X(-,[0]) \to GMap(E_\bullet, X(-,\bullet))$ is an equivalence of $G$-spaces.

Here $E_\bullet$ is included with trivial $G$-action, and $GMap$ denotes the $Top_G$-enriched homset of simplicial $G$-spaces.

Comparison:

These two notions are indeed equivalent. We see this as follows.

1. Since limits in a presheaf category are computed levelwise, the Segal conditions in the two cases are equivalent.

2. $GMap(E_\bullet, X(-,\bullet))(G/H) = Map(E_\bullet, X(G/H,\bullet))$ because for each $[n]$, $E(G/H,[n])$ is constant as a function of $G/H$ (for me, this implication requires a short calculation). Thus the univalence conditions are also equivalent.

Enriched categories? So I guess that a $G$-enriched category is an internal category to $GTop$ with trivial $G$-action on the space of objects? This should translate to a parameterized category where the "core" -- the maximal subfibration which is fibered in groupoids -- is constant?

It seems that the answer to the first part of the revised question is yes: categories parameterized over $O_G$ are the same as categories internal to $Top_G$.

Let's think through this carefully. A category object should satisfy Segal conditions and a univalence (aka completeness in the sense of Rezk) condition. Admittedly the latter might not be entirely standardized for internal categories, but let's pick a formulation and run with it:

• A category parameterized over $O_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ which is, levelwise in $O_G$, a complete Segal space. So we have

1. Segal conditions: $X(G/H,[n]) \to X(G/H,[1]) \times_{X(G/H,[0])} \dots \times_{X(G/H,[0])} X(G/H,[1])$ is an equivalence.

2. Univalence: $X(G/H,[0]) \to Map(E_\bullet,X(G/H,\bullet))$ is an equivalence of spaces.

Here $E_\bullet$ is the (nerve of the) walking isomorphism, and $Map$ the $Top$-enriched homset of simplicial spaces.

• A category internal to $Top_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ satisfying

1. Segal conditions: $X(-,[n]) \to X(-,[1]) \times_{X(G/H,[0])} \dots \times_{X(-,[0])} X(-,[1])$ is an equivalence.

2. Univalence: $X(-,[0]) \to GMap(E_\bullet, X(-,\bullet))$ is an equivalence of $G$-spaces.

Here $E_\bullet$ is included with trivial $G$-action, and $GMap$ denotes the $Top_G$-enriched homset of simplicial $G$-spaces.

Comparison:

These two notions are indeed equivalent. We see this as follows.

1. Since limits in a presheaf category are computed levelwise, the Segal conditions in the two cases are equivalent.

2. $GMap(E_\bullet, X(-,\bullet))(G/H) = Map(E_\bullet, X(G/H,\bullet))$ because for each $[n]$, $E(G/H,[n])$ is constant as a function of $G/H$ (for me, this implication requires a short calculation). Thus the univalence conditions are also equivalent.

Enriched categories? So I guess that a $G$-enriched category is an internal category to $GTop$ with trivial $G$-action on the space of objects? This should translate to a parameterized category where the "core" -- the maximal subfibration which is fibered in groupoids -- is constant?

Let's think through this carefully. A category object should satisfy Segal conditions and a univalence (aka completeness in the sense of Rezk) condition. Admittedly the latter might not be entirely standardized for internal categories, but let's pick a formulation and run with it:

• A category parameterized over $O_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ which is, levelwise in $O_G$, a complete Segal space. So we have

1. Segal conditions: $X(G/H,[n]) \to X(G/H,[1]) \times_{X(G/H,[0])} \dots \times_{X(G/H,[0])} X(G/H,[1])$ is an equivalence.

2. Univalence: $X(G/H,[0]) \to Map(E_\bullet,X(G/H,\bullet))$ is an equivalence of spaces.

Here $E_\bullet$ is the (nerve of the) walking isomorphism, and $Map$ the $Top$-enriched homset of simplicial spaces.

• A category internal to $Top_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ satisfying

1. Segal conditions: $X(-,[n]) \to X(-,[1]) \times_{X(G/H,[0])} \dots \times_{X(-,[0])} X(-,[1])$ is an equivalence.

2. Univalence: $X(-,[0]) \to GMap(E_\bullet, X(-,\bullet))$ is an equivalence of $G$-spaces.

Here $E_\bullet$ is included with trivial $G$-action, and $GMap$ denotes the $Top_G$-enriched homset of simplicial $G$-spaces.

Comparison:

These two notions are indeed equivalent. We see this as follows.

1. Since limits in a presheaf category are computed levelwise, the Segal conditions in the two cases are equivalent.

2. $GMap(E_\bullet, X(-,\bullet))(G/H) = Map(E_\bullet, X(G/H,\bullet))$ because for each $[n]$, $E(G/H,[n])$ is constant as a function of $G/H$ (for me, this statement requires a short calculation). Thus the univalence conditions are also equivalent.

Enriched categories? So I guess that a $G$-enriched category is an internal category to $GTop$ with trivial $G$-action on the space of objects? This should translate to a parameterized category where the "core" -- the maximal subfibration which is fibered in groupoids -- is constant?

It seems that the answer to the revised question is yes: categories parameterized over $O_G$ are the same as categories internal to $Top_G$.

Let's think through this carefully. A category object should satisfy Segal conditions and a univalence (aka completeness in the sense of Rezk) condition. Admittedly the latter might not be entirely standardized for internal categories, but let's pick a formulation and run with it:

• A category parameterized over $O_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ which is, levelwise in $O_G$, a complete Segal space. So we have

1. Segal conditions: $X(G/H,[n]) \to X(G/H,[1]) \times_{X(G/H,[0])} \dots \times_{X(G/H,[0])} X(G/H,[1])$ is an equivalence.

2. Univalence: $X(G/H,[0]) \to Map(E_\bullet,X(G/H,\bullet))$ is an equivalence of spaces.

Here $E_\bullet$ is the (nerve of the) walking isomorphism, and $Map$ the $Top$-enriched homset of simplicial spaces.

• A category internal to $Top_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ satisfying

1. Segal conditions: $X(-,[n]) \to X(-,[1]) \times_{X(G/H,[0])} \dots \times_{X(-,[0])} X(-,[1])$ is an equivalence.

2. Univalence: $X(-,[0]) \to GMap(E_\bullet, X(-,\bullet))$ is an equivalence of $G$-spaces.

Here $E_\bullet$ is included with trivial $G$-action, and $GMap$ denotes the $Top_G$-enriched homset of simplicial $G$-spaces.

Comparison:

These two notions are indeed equivalent. We see this as follows.

1. Since limits in a presheaf category are computed levelwise, the Segal conditions in the two cases are equivalent.

2. $GMap(E_\bullet, X(-,\bullet))(G/H) = Map(E_\bullet, X(G/H,\bullet))$ because for each $[n]$, $E(G/H,[n])$ is constant as a function of $G/H$ (for me, this implication requires a short calculation). Thus the univalence conditions are also equivalent.

Enriched categories? So I guess that a $G$-enriched category is an internal category to $GTop$ with trivial $G$-action on the space of objects? This should translate to a parameterized category where the "core" -- the maximal subfibration which is fibered in groupoids -- is constant?