I personally think of a reference frame as being a velocity and position in space from which to observe things, and a coordinate system as being the measuring system by which we locate things at other positions. It matches the general usage of the terms better.
Essentially, the reference frame describes here, while the coordinate system describes everywhere else with respect to here.
Note phrases like this (from IsaacPhysics.org) that are quite common:
The different observations occur because the two observers are in different frames of reference.
You can't be in two different coordinate systems. You can be at different locations, or have different velocities, in a single coordinate system. You can use different coordinate systems. But both observers are in locations, at velocities, described by both coordinate systems.
That snippet is followed by the definition:
A frame of reference is a set of coordinates that can be used to
determine positions and velocities of objects in that frame;
different frames of reference move relative to one another.
This essentially defines a reference frame as a coordinate system, but that doesn't make sense in context. It says a reference frame is something that can move. A coordinate system spans all of spacetime, but something that moves needs a finite space it occupies so we can locate it and therefore determine it's moving. The thing that's moving is the origin of the coordinate system.
A reference frame is something that moves. The origin of a coordinate system moves.
An object can exist in one reference frame while not existing in another. Two coordinate system origins can be at different locations, moving at different speeds.
The usages of "frame of reference" in the quotes above are satisfied by calling it the origin of a coordinate system. But they're not satisfied by calling it a coordinate system, since an object can't exist in one coordinate system but not the other, and coordinate systems don't really have locations as they span everywhere.
The origin of a coordinate system is a position and velocity in space that we use to calculate how other objects would look from said origin, leading to the simple definition I gave above:
A reference frame is a velocity and position in space from which to observe things.
Side note 1: we don't have to use the origin. We could substitute "origin" with "specific location", since all the locations within a coordinate system can move like the origin, and an object can be in one location in one coordinate system and not be at a specific location in another coordinate system. But it's generally simplest to use coordinate systems whose origins are located at the point from which we want to view things.
Side note 2: "position" here includes rotational orientation as well as linear position.