Note: I am sorry for posting an answer. It should be a Comment, but I can't: surprinsigly, I need 50 karma to do that... even for an answer to my question.

@Bole, first of all, THANK YOU VERY MUCH!!! It works!!!

Second, what you explained is entirely logical... but I assumed Odoo would better understand the inheritance. Why I have to assume I have to "redeclare" parent fields??? Please, it's an inheritance!!! If I have a model, and I have to inherit from it... what is the sense of the inheritance if I have to redeclare all fields??? (Assuming all parent fields are relations to other models).

Third, OK, I understand developers must know how works the platform they are going to develop on... but how to know how Odoo works? Where is the developer documentation?  

Fourth, I am sorry. I went to mark your reply as the Accepted one, but I need more Karma... 

Well... 
class implementation_project(osv.Model):
    _name = 'implementation.project'
    _inherit = 'project.project'

What you did is, you created a completley new object 'implementation.project' wich inherited all fields from original class...
and then you added some new custom fields...
But, what you did NOT do is.. change some of inherited fields... one of them is :
'type_ids': fields.many2many('project.task.type', 'project_task_type_rel', 'project_id', 'type_id', 'Tasks Stages', states={'close':[('readonly',True)], 'cancelled':[('readonly',True)]}),

SO now you have your new model, with task type related to old model.. 
what you should do is override that field like:
'type_ids': fields.many2many('project.task.type', 'implementation_project_task_type_rel', 'project_id', 'type_id', 'Tasks Stages', states={'close':[('readonly',True)], 'cancelled':[('readonly',True)]}), 
in order to establish a new relatin table between your new created model and old task_type model... 

Now cretaing record should work just fine.. 

hope it helps : )

p.s. 
odoo is nice platform for rapid development of businees apps... but, in order to use it you have to firs know how it works...
in other words RTFM :)

Odoo has a couple of way that you can inherit an object from.  One way is how you did, which is using the _inherit attribute with a string value that matches the name of the model that you want to inherit.  This way Odoo will extend the original model.  The class name bears no "meaning" to the ORM model.  The _name attribute, however may change the _name attribute of the original model (can't say for sure as I haven't really check it out).  When you create a record, you need to specify all the values of the necessary fields.  All values will be written on the same table.

The second way is to use _inherits = {'model.name': 'field_name'}.  This is done, e.g. for product.product in which it inherit product.template (check out the odoo/odoo/addons/product/product.py file).  This is a creation of another class that inherits a different parent class.  The sub class will have 'field_name' but all the fields/columns that belongs to the parent class will be written in the parent's class table.  When you create a record, you need to specify all the values of the necessary fields.  Values belonging to parent class will be written into the parent's table and values beloging to child class will be written in the child's table.

The third way, usually used for mixin is to use _inherit = ['mixin.model'].  You can also checkout product.product as it also inherits 'mail.thread'.  As it is just extension of capability, it adds no additional fields.