From the English rules published by the publisher here:
The Beetle, like the Queen Bee, moves only one space per turn. Unlike any other creature though, it can also move on top of the Hive.
There are further words on what the Beetle does when on top of the Hive, but they're not relevant here.
Freedom to move
The creatures can only move in a sliding movement. If a piece is surrounded to the point that it can no longer physically slide out of its position, it may not be moved. The only exceptions are the Grasshopper, which can jump into or out of a space, and the Beetle, which can also climb up or down.
The key discussion point here is the word "also" in "the Beetle, which can also climb up or down". "Also" meaning "as well as", but "as well as" what? The Beetle can do some things as well as being able to do what the Grasshopper can do, i.e. jump into or out of a space as well as climbing up or down? Or the Beetle can climb up or down as well as being able to slide?
My native English speaker interpretation of that wording is that the Beetle can jump into or out of a space as well as being able to slide. The Beetle can climb up or down, but that doesn't imply it can do both in one turn. Therefore the move in yellow would be disallowed.
From the same rules, under the Moving section it states:
**If a piece is the only connection between two part so the Hive, it may not be moved. (See 'One Hive rule' p9)
That rule on p9 stating:
One Hive rule
The pieces in play must be linked at all times. At no time can you leave a piece stranded (not joined to the Hive) or separate the Hive in two.
In an example the One Hive rule has the text:
Moving the black Queen Bee to a position where it re-links the Hive is also an illegal move, as the Hive is left unlinked while the piece is in transit.
I see no difference between your second diagram and the example move: you're taking a piece that is the only link, and moving it to an adjacent position. While moving it appears that the moving piece is not considered to be connected to the Hive. Therefore your red move would be disallowed.