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(),

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

@@ -219,7 +219,7 @@ pub fn action(input: &Value, _args: &Arguments, span: Span) -> Value {
         Value::Error { .. } => input.clone(),
         other => Value::error(
             ShellError::OnlySupportsThisInputType {
-                exp_input_type: "integer, filesize, string, date, duration, binary or bool".into(),
+                exp_input_type: "int, filesize, string, date, duration, binary, or bool".into(),
                 wrong_type: other.get_type().to_string(),
                 dst_span: span,
                 src_span: other.span(),

crates/nu-cmd-extra/src/extra/bits/not.rs

@@ -158,7 +158,7 @@ fn operate(value: Value, head: Span, signed: bool, number_size: NumberBytes) ->
             Value::Error { .. } => 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(),

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

@@ -22,16 +22,12 @@ impl Command for BitsOr {
                     Type::List(Box::new(Type::Int)),
                 ),
             ])
-            .required(
-                "target",
-                SyntaxShape::Int,
-                "target integer to perform bit or",
-            )
+            .required("target", SyntaxShape::Int, "target int to perform bit or")
             .category(Category::Bits)
     }
 
     fn usage(&self) -> &str {
-        "Performs bitwise or for integers."
+        "Performs bitwise or 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(),

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

@@ -41,7 +41,7 @@ impl Command for BitsRol {
     }
 
     fn usage(&self) -> &str {
-        "Bitwise rotate left for integers."
+        "Bitwise rotate left for ints."
     }
 
     fn search_terms(&self) -> Vec<&str> {
@@ -145,7 +145,7 @@ fn operate(value: Value, bits: usize, head: Span, signed: bool, number_size: Num
         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(),

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

@@ -41,7 +41,7 @@ impl Command for BitsRor {
     }
 
     fn usage(&self) -> &str {
-        "Bitwise rotate right for integers."
+        "Bitwise rotate right for ints."
     }
 
     fn search_terms(&self) -> Vec<&str> {
@@ -149,7 +149,7 @@ fn operate(value: Value, bits: usize, head: Span, signed: bool, number_size: Num
         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(),

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

@@ -41,7 +41,7 @@ impl Command for BitsShl {
     }
 
     fn usage(&self) -> &str {
-        "Bitwise shift left for integers."
+        "Bitwise shift left for ints."
     }
 
     fn search_terms(&self) -> Vec<&str> {
@@ -169,7 +169,7 @@ fn operate(value: Value, bits: usize, head: Span, signed: bool, number_size: Num
         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(),

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

@@ -41,7 +41,7 @@ impl Command for BitsShr {
     }
 
     fn usage(&self) -> &str {
-        "Bitwise shift right for integers."
+        "Bitwise shift right for ints."
     }
 
     fn search_terms(&self) -> Vec<&str> {
@@ -159,7 +159,7 @@ fn operate(value: Value, bits: usize, head: Span, signed: bool, number_size: Num
         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(),

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

@@ -22,16 +22,12 @@ impl Command for BitsXor {
                     Type::List(Box::new(Type::Int)),
                 ),
             ])
-            .required(
-                "target",
-                SyntaxShape::Int,
-                "target integer to perform bit xor",
-            )
+            .required("target", SyntaxShape::Int, "target int to perform bit xor")
             .category(Category::Bits)
     }
 
     fn usage(&self) -> &str {
-        "Performs bitwise xor for integers."
+        "Performs bitwise xor for ints."
     }
 
     fn search_terms(&self) -> Vec<&str> {
@@ -84,7 +80,7 @@ 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(),

crates/nu-cmd-extra/src/extra/conversions/fmt.rs

@@ -88,7 +88,7 @@ fn action(input: &Value, _args: &CellPathOnlyArgs, span: Span) -> Value {
         Value::Error { .. } => input.clone(),
         other => Value::error(
             ShellError::OnlySupportsThisInputType {
-                exp_input_type: "float , integer or filesize".into(),
+                exp_input_type: "float, int, or filesize".into(),
                 wrong_type: other.get_type().to_string(),
                 dst_span: span,
                 src_span: other.span(),