fix(install): use install.ps1 immediately + honest PATH guidance

The persisted User PATH only reaches newly-started processes, so the old
"restart your shell" advice was wrong — a sign-out/in was actually needed
(observed on Windows 11). Update the current session's $env:Path as well
so the command works right away in the same window, and reword the notice.

Also refresh the header: verified on ARM64 Windows 11 under Windows
PowerShell 5.1 and PowerShell 7.6 against the live v0.2.0 release.

scripts/install.ps1

@@ -24,9 +24,11 @@
     %LOCALAPPDATA%\Programs\rdbms-playground.
 
 .NOTES
-    Written but NOT tested on Windows from this environment (no PowerShell
-    here) — validate on a real Windows host. The verified sibling is
-    install.sh (Linux/macOS).
+    Verified end-to-end on ARM64 Windows 11 under both Windows PowerShell 5.1
+    and PowerShell 7.6, against the live v0.2.0 release. The x86_64 branch is
+    symmetric (env-based arch detection + a confirmed matching release asset)
+    but has not been run directly. The sibling installer is install.sh
+    (Linux/macOS).
 #>
 [CmdletBinding()]
 param(
@@ -98,14 +100,22 @@ try {
     Move-Item -Path $exe -Destination $dest -Force
     Write-Host "installed $Bin $Version -> $dest"
 
-    # Add the install dir to the user PATH (persistent) if it's not there.
+    # Persist the install dir on the user PATH (for future shells) if missing.
     $userPath = [Environment]::GetEnvironmentVariable('Path', 'User')
     if (-not $userPath) { $userPath = '' }
     if (($userPath -split ';') -notcontains $InstallDir) {
         $newPath = if ($userPath) { "$userPath;$InstallDir" } else { $InstallDir }
         [Environment]::SetEnvironmentVariable('Path', $newPath, 'User')
-        Write-Host "added $InstallDir to your user PATH — restart your shell to pick it up"
     }
+    # Also update THIS session's PATH so the command works immediately. The
+    # persisted change only reaches newly-started processes; an already-running
+    # shell (and, depending on how the terminal inherited its environment, even
+    # a freshly-opened one) won't see it until the next sign-out/in.
+    if (($env:Path -split ';') -notcontains $InstallDir) {
+        $env:Path = "$env:Path;$InstallDir"
+    }
+    Write-Host "added $InstallDir to your PATH — '$Bin' works in this window now."
+    Write-Host "for shells already open elsewhere, sign out and back in (or open a fresh one)."
 }
 finally {
     Remove-Item -Path $tmp -Recurse -Force -ErrorAction SilentlyContinue