It's the same.

First, you can notice that if you have a set of compact objects $F$ in a triangulated category $S$ such that there's no proper triangulated, coproduct-closed subcategory containing $F$, then minimal thick subcategory containing $F$ is exactly subcategory of compact objects. (It's more or less a consequence of triangulated Brown representability theorem.)

Then if $T$ is $D(R{\operatorname{-mod}})$, $R$ is a compact object in it (since homs from it are zeroth cohomology); smallest thick subcategory containing $R$ is the subcategory of perfect complexes, so by observation above it is the subcategory of compact objects.

If you are not satisfied with my brief explanation, this is written more carefully in Neeman's book on triangulated categories.

It's the same.

First, you can notice that if you have a set of compact objects $F$ in a triangulated category $S$ such that there's no proper triangulated, coproduct-closed subcategory containing $F$, the minimal thick subcategory containing $F$ is exactly the subcategory of compact objects. (It's more or less a consequence of the triangulated Brown representability theorem.)

Then if $T$ is $D(R{\operatorname{-mod}})$, $R$ is a compact object in it (since homs from it are zeroth cohomology); the smallest thick subcategory containing $R$ is the subcategory of perfect complexes, so by the observation above it is the subcategory of compact objects.

If you are not satisfied with my brief explanation, this is written more carefully in Neeman's book on triangulated categories.

It's the same.

First, you can notice that if you have a set of compact objects $F$ in a triangulated category $S$ such that there's no proper triangulated, coproduct-closed subcategory containing $F$, then minimal thick subcategory containing $F$ is exactly subcategory of compact objects. (It's more or less a consequence of triangulated Brown representability theorem.)

Then if $T$ is $D(R-mod)$, $R$ is a compact object in it (since homs from it are zeroth cohomology); smallest thick subcategory containing $R$ is the subcategory of perfect complexes, so by observation above it is the subcategory of compact objects.

If you are not satisfied with my brief explanation, this is written more carefully in Neeman's book on triangulated categories.

It's the same.

First, you can notice that if you have a set of compact objects $F$ in a triangulated category $S$ such that there's no proper triangulated, coproduct-closed subcategory containing $F$, then minimal thick subcategory containing $F$ is exactly subcategory of compact objects. (It's more or less a consequence of triangulated Brown representability theorem.)

Then if $T$ is $D(R{\operatorname{-mod}})$, $R$ is a compact object in it (since homs from it are zeroth cohomology); smallest thick subcategory containing $R$ is the subcategory of perfect complexes, so by observation above it is the subcategory of compact objects.

If you are not satisfied with my brief explanation, this is written more carefully in Neeman's book on triangulated categories.