What is the difference between clinical and non-clinical depression, and is there a term for different severity of the bipolar disorder?
I think the main question here is:


  Is there a term…which describes a bipolar disorder of lesser severity[?]


Yes. There are a couple ways to think about this, but you’re clearly accustomed to differentiating the manic and depressive phases of bipolar disorder, so I’ll start with that. 

There is mania (excerpted from DSM 5*):


  A. A distinct period of abnormally and persistently elevated, expansive, or irritable mood and abnormally and persistently increased goal-directed activity or energy, lasting at least 1 week and present most of the day, nearly every day...
  
  B. During the period of mood disturbance and increased energy or activity, three (or more) of the following symptoms (four if the mood is only irritable) are present to a significant degree and represent a noticeable change from usual behavior: 
  
  
  Inflated self-esteem or grandiosity.
  Decreased need for sleep 
  More talkative than usual or pressure to keep talking.
  Flight of ideas or subjective experience that thoughts are racing.
  Distractibility 
  Increase in goal-directed activity 
  Excessive involvement in activities that have a high potential for painful consequences 
  
  
  C. The mood disturbance is sufficiently severe to cause marked impairment in social or occupational functioning....


And there is hypomania:


  A. A distinct period of abnormally and persistently elevated, expansive, or irritable mood and abnormally and persistently increased activity or energy, lasting at least 4 consecutive days and present most of the day, nearly every day. ….
  
  B. [This criteria is the same as for mania.]
  ....
  E. The episode is not severe enough to cause marked impairment in social or occupational functioning or to necessitate hospitalization.


I have bolded the main differences between the two definitions. You can see that hypomania is the term for the less severe form of the manic phase of bipolar depression. 

Neither mania nor hypomania is likely to occur in the absence of depressive symptoms, however. Thus, there are two basic phenotypes of bipolar disorder (summarized from DSM 5*):

Bipolar I disorder: Patient experiences mania; depressive periods may not qualify as Major Depressive Disorder. 

Bipolar II disorder: Major Depressive episodes are required; manic phase may be hypomanic. (This is classically considered a less severe syndrome, but this notion has been challenged in recent years.)

And finally, one more term for you:  

Cyclothymic disorder: patient experiences both hypomanic and depressive periods without ever fulfilling the criteria for an episode of mania, hypomania, or major depression.

(Your other question you seem to already know the answer to, but since it's in the title.....  Nonclinical depression refers to something on the spectrum of normal (i.e. not pathological), such as situational mood disturbance. A related concept, subclinical depression is a loose term to describe depressive symptoms that don't meet criteria for Major Depressive Disorder..)



*American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders (5th ed.). Arlington, VA: American Psychiatric Publishing.