forms do not have a 'name' attribute. They have an 'ID' attribute and id's a unique, can only be used once.
Your using javascript to submit the form
Code:
<a href="javascript:document.submitForm.submit()">
But you don't show your script. Why not just use the submit button?
Code:
<table >
<thead >
<tr>
<th>
Heading 1
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<form id="form1" action="mylink.php" method="post">
<button type="submit" value="PUSH">Submit</button>First
</form>
</td>
</tr>
<tr>
<td>
<form id="form2" action="mylink.php" method="post">
<button type="submit" value="PUSH">Submit</button>Second
</form>
</td>
</tr>
</tbody>
</table>
Will activate mylink.php, but wont do anything beyond that.