Let’s say I try to assign a string to a strongly typed integer variable:

This works, as PowerShell is able to cast the string '1' to an integer.

Now things are different if I attempt the same with a strongly typed boolean variable:

I find this confusing, as PowerShell at the same time allows an explicit cast from string to boolean:

Does anyone know the reason for this seemingly inconsistent behavior?

Practical advice:

In most cases I would argue the following approach, rather than typed variables: Convert your values to the target type prior to assignment, and then let the type system infer the variable type:

The problem at hand:

(This is not an authoritative answer, but a best-guess)

Briefly mentioned in the about_Variables help file:

TYPES OF VARIABLES

You can store any type of object in a variable, […]

Windows PowerShell variables are “loosely typed,” which means that
they are not limited to a particular type of object. […]

[… section about value-inferred types bla bla bla …]

You can use a type attribute and cast notation to ensure that a
variable can contain only objects of the specified type or objects
that can be converted to that type. If you try to assign a value
of another type, Windows PowerShell tries to convert the value to
its type. If it cannot, the assignment statement fails.

To use cast notation, enter a type name, enclosed in brackets, before
the variable name (on the left side of the assignment statement).

Although “type attribute” and the distinct constraint that this applies only during assignment is used nowhere else in the documentation, but (to me at least) indicates that this is a special case of explicit casting.

What does this imply?

When you add an explicit cast notation to the variable, during assignment, as described above, PowerShell adds an ArgumentTypeConverterAttribute to the variable, and PowerShell’s type conversion magic is suddenly overridden by a type-specific Transform() method:

If we repeat this experiment with a boolean type, you can see how the ArgumentTypeConverterAttribute transform is much more restrictive than the normal conversion “magic”:

Again in this type of conversion Boolean parameters accept only Boolean values and numbers, such as $True, $False, 1 or 0., whereas powershell itself generally interprets any non-empty, non-zero or non-null value to be $true when implicitly converted to [bool].

in other words:

Cast notations are context-sensitive:

Even though they look just the same