Use int type name consistently (#10579)

# Description
When referring to the type use `int` consistently. Only when referring
to the concept of integer numbers use `integer`.

- Fix `random integer` to `random int` tests
  - Forgot in #10520
- Use int instead of integer in error messages
- Use int type name in bits commands
- Fix messages in `for` examples
- Use int typename in `into` commands
- Use int typename in rest of commands
- Report errors in `nu-protocol` with int typename

Work for #10332 

# User-Facing Changes
User errorrs should now use `int` so you can easily find the necessary
commands or type annotations.

# Tests + Formatting
Only two tests found that needed updating

crates/nu-cmd-extra/src/extra/bits/and.rs

@@ -22,16 +22,12 @@ impl Command for BitsAnd {
                     Type::List(Box::new(Type::Int)),
                 ),
             ])
-            .required(
-                "target",
-                SyntaxShape::Int,
-                "target integer to perform bit and",
-            )
+            .required("target", SyntaxShape::Int, "target int to perform bit and")
             .category(Category::Bits)
     }
 
     fn usage(&self) -> &str {
-        "Performs bitwise and for integers."
+        "Performs bitwise and for ints."
     }
 
     fn search_terms(&self) -> Vec<&str> {
@@ -85,7 +81,7 @@ fn operate(value: Value, target: i64, head: Span) -> Value {
         Value::Error { .. } => value,
         other => Value::error(
             ShellError::OnlySupportsThisInputType {
-                exp_input_type: "integer".into(),
+                exp_input_type: "int".into(),
                 wrong_type: other.get_type().to_string(),
                 dst_span: head,
                 src_span: other.span(),