Multiple form authentication in the same domain - php

Question

I have a website with two form authentication in different pages, have different input name and link to different pages . The problem is that when I save my authentication to a browser (chrome) of a form , the browser fill in the fields with the same data in the other form . How is it possible?
First form
<form action="" method="POST" enctype="multipart/form-data">
<div class="form-group">
<label for="exampleInputEmail1">Email</label>
<input type="email" name="private_email" class="form-control" id="email1" value="" placeholder="Enter email" required>
</div>
<div class="form-group">
<label for="exampleInputPassword1">Password</label>
<input type="password" class="form-control" name="private_password" value="" id="password1" placeholder="Password" required>
</div>
<input type="submit" name="login" class="btn btn-default" value="Login">
</form>
Second Form (It is a form of a cms)
<form action="http://escuolainsieme.it/shop/login" method="post" id="login_form" class="box">
<h3 class="page-subheading">Sei già registrato?</h3>
<div class="form_content clearfix">
<div class="form-group form-ok">
<label for="email">Indirizzo email</label>
<input class="is_required validate account_input form-control" data-validate="isEmail" type="text" id="email" name="email" value="">
</div>
<div class="form-group">
<label for="passwd">Password</label>
<span><input class="is_required validate account_input form-control" type="password" data-validate="isPasswd" id="passwd" name="passwd" value=""></span>
</div>
<p class="lost_password form-group">Hai dimenticato la password?</p>
<p class="submit">
<input type="hidden" class="hidden" name="back" value="my-account"> <button type="submit" id="SubmitLogin" name="SubmitLogin" class="button btn btn-default button-medium">
<span>
<i class="icon-lock left"></i>
Entra
</span>
</button>
</p>
</div>
</form>
Login.php
<?php session_start(); // Starting Session
$error = ''; // Variable To Store Error Message
if (isset($_POST['private_login'])) {
if (empty($_POST['private_email']) || empty($_POST['private_password'])) {
$error = "<div class='alert alert-danger'>Compila tutti i campi</div>";
} else {
$email = mysqli_real_escape_string(conn(), $_POST['private_email']);
$password = mysqli_real_escape_string(conn(), $_POST['private_password']);
$cls_utente = new utente();
if ($cls_utente->check_user($email, $password) == 1) {
$_SESSION['login_user'] = $email; // Initializing Session
$_SESSION['is_logged'] = true;
} else {
$error .= "<div class='alert alert-danger'>Email o password errati</div>";
}
}}?>

Answer 1

You can try using autocomplete="false" or autocomplete="off" to disable it, not sure if it will work but give it a try.
As far as i am concerned it's not possible to make the browser 'realize' that they are different forms and they should not be auto-filled with the same data.
Have a look at these 2 answers, answer1 answer2 for more information how browser detects the forms.

Answer 2

You must add another one column in your user table, example if you add type column the value set default super admin,moderator, user. Now you can check the login authentication with this column.If user-name and password and type is equal redirect to particular page.so you can redirect different page depends upon the user type..