Hello Magen,

Here is Posibillity

1) Because of In Both Email Template Same Class Used May be that was creating Problem To Chose Right one During Send Mail.

Solution ::

You can Define Your Own Custom Sending Mail Function On Button and Send Mail using That, and Use Template Which you want to send.

@api.modeldef send_mail_to_partner(self):    
    picking_ids = self.env['stock.picking'].search([])    
    for pick in picking_ids:        
        ir_model_data = self.env['ir.model.data']        
        try:            
            template_id = email_temp.id        
        except ValueError:            
            template_id = False        
        temp = self.env.ref('MODULE_NAME.EMAIL_TEMPLATE_NAME')        
        self.env['mail.template'].browse(temp.id).send_mail(pick.id, force_send=True, raise_exception=True)

<button name="send_mail_to_partner" type="object" string="Send Mail" class="oe_highlight"/>