There is no standard definition of the local vertical, local horizontal frame. There are four key choices:

1. Is local vertical (aka radial) toward or away from the center of the Earth?
2. Is another axis along or against the angular velocity vector ($\vec r \times \vec v$)?
3. Should we use a right-handed or left-handed coordinate system?
4. Which axis is which regarding the xyz order of the axes?

Item (3) has been pretty much standardized in favor of a right-handed coordinate system. The remaining three have not been standardized. When reading about Hill's equations, also known as the Clohessy-Wiltshire equations, it is very important to understand the coordinate system used in that reference and to be able to transform the equations to your chosen form.

The first set of equations in the question are using $\hat x$ as pointing away from the center of the Earth, $\hat z$ as pointing along the angular velocity vector, and $\hat y$ as completing a right-handed coordinate system. The second set of equations appears to be using $\hat z$ as pointing toward the center of the Earth as opposed to radially away. Caveat reader!

There is no standard definition of the local vertical, local horizontal frame. There are four key choices:

1. Is local vertical (aka radial) toward or away from the center of the Earth?
2. Is another axis along or against the angular velocity vector ($\vec r \times \vec v$)?
3. Should we use a right-handed or left-handed coordinate system?
4. Which axis is which regarding the xyz order of the axes?

Item (3) has been pretty much standardized in favor of a right-handed coordinate system. The remaining three have not been standardized. When reading about Hill's equations, also known as the Clohessy-Wiltshire equations, it is very important to understand the coordinate system used in that reference and to be able to transform the equations to your chosen form.

The first set of equations in the question appears to be using $\hat x$ as pointing away from the center of the Earth, $\hat z$ as pointing along the angular velocity vector, and $\hat y$ as completing a right-handed coordinate system. The second set of equations appears to be using $\hat z$ as pointing toward the center of the Earth as opposed to radially away.

Every writeup I have seen have uniformly made choices for items 1, 2, and 4 such that the third axis more or less points along rather than against the velocity vector. (The CW equations can be used for orbits that are very slightly elliptical, which is why I wrote "more or less".) The choices of items 1, 2, and 4 are however not standardized.

Caveat reader!

There is no standard definition of the local vertical, local horizontal frame. There are four key choices:

1. Is local vertical (aka radial) toward or away from the center of the Earth?
2. Is another axis along or against the angular velocity vector ($\vec r \times \vec v$)?
3. Should we use a right-handed or left-handed coordinate system?
4. Which axis is which regarding the xyz order of the axes?

Item (3) has been pretty much standardized in favor of a right-handed coordinate system. The remaining three have not been standardized. When reading about Hill's equations, also known as the Clohessy-Wiltshire equations, it is very important to understand the coordinate system used in that reference and to be able to transform the equations to your chosen form.

There is no standard definition of the local vertical, local horizontal frame. There are four key choices:

1. Is local vertical (aka radial) toward or away from the center of the Earth?
2. Is another axis along or against the angular velocity vector ($\vec r \times \vec v$)?
3. Should we use a right-handed or left-handed coordinate system?
4. Which axis is which regarding the xyz order of the axes?

Item (3) has been pretty much standardized in favor of a right-handed coordinate system. The remaining three have not been standardized. When reading about Hill's equations, also known as the Clohessy-Wiltshire equations, it is very important to understand the coordinate system used in that reference and to be able to transform the equations to your chosen form.

The first set of equations in the question are using $\hat x$ as pointing away from the center of the Earth, $\hat z$ as pointing along the angular velocity vector, and $\hat y$ as completing a right-handed coordinate system. The second set of equations appears to be using $\hat z$ as pointing toward the center of the Earth as opposed to radially away. Caveat reader!