What verb would sound more like a legal term (to be used in documents) if one wants to write that he fires a worker from director's position? To fire? To terminate position?
It depends on the circumstances in which it is happening. If it is a sudden 'summary dismissal' for misconduct, probably 'dismiss', is the right term.
If it is redundancy (what Americans call 'lay off') then one of those terms might be best used.
If it is dismissal with notice etc for poor job performance, then 'severance of contract', 'termination', might be appropriate. If this happens in the UK, it is likely to be subject to intense legal scrutiny, if the former employee decides to go to an Industrial Tribunal.
For a senior position most employers are nowadays advised to come to an arrangement by consent, and agree a termination package. The employee in question would be given a termination agreement for signature (known in the UK as a 'compromise agreement' on which he or she would be advised by any responsible employer to take independent legal advice).
In America, employment law is heavily weighted toward 'hiring practices'. You have to be desperately concerned about the manner in which you hire people so as to show no discrimination. In Europe it is weighted far more toward 'firing practices'. Sacking a senior person can involve negotiating a legal minefield.
In US usage, it is common to overuse the term to fire. Correctly construed, firing = termination with cause (i.e., the employee has violated the employee handbook, performed poorly over time, or broken a law). However, as applied especially to terminating executives, cause is virtually irrelevant, hence the overuse of the term to fire (e.g., as we see on the television show The Apprentice). By comparison, termination without cause = layoff (which by definition is a decision that considers the economics of maintaining the affected job position or number of employees in a given position). In US practice, a layoff under this definition requires reporting to the state, for purposes of recalculating the employer's unemployment insurance premium (because the probability of a layoff with that employer has just increased if one uses past statistics to predict future likelihoods). For this reason, the difference between a layoff and a firing in the US is actually quite substantive, especially for wage-grade (FLSA-exempt) employees. The term dismissal in US usage is synonymous with termination but has a gentler quality. For even greater gentleness, one can say that one has let go of the employee. Because a reference to termination (or dismissal, or letting go) by itself leaves cause unspecified, one can use it in reference to a firing if one wishes to be vague on the matter.