Try clearing the browser cookies. Logon credentials are often stored in browser cookies, which you can delete by a call to
CookieManager.getInstance().removeAllCookie();
Update:
CookieManager.getInstance().removeAllCookie() was deprecated in Android SDK 21 (Lollipop), and was replaced by CookieManager.getInstance().removeAllCookies(ValueCallback callback). As of November 2018, removeAllCookie() stills works as specified. Also, in the newer method removeAllCookies(ValueCallback callback), the callback function can be null if you don't want to be notified about removed cookies. Thus, it's safe to use
CookieManager.getInstance().removeAllCookies(null)
as a direct replacement for CookieManager.getInstance().removeAllCookie() when targeting devices after Android SDK 21.
CookieManager documentation is here.