What does not matter:
• I am based out of Houston
• paid in dollars
• into my UK bank account
What does matter:
• it is for an American oil Company
• I am training
• I also undertake some classroom based courses
• I also have spent time offshore on rigs in Nigeria and Korea
• I only have an esta
You have not mentioned:
• whether your travel is for 90 days or less
• whether the training is of you or by you
A UK citizen (amongst others!) must obtain a visa to travel to the United States if traveling there for a purpose other than short-term tourism or business. It seems even your employer is not claiming the former motive, so it is the latter that is critical:
• consult with business associates
• attend a scientific, educational, professional, or business convention or conference
• attend short-term training (you may not be paid by any source in the United States with the exception of expenses incidental to your stay)
• negotiate a contract
Some of your activities do seem to comply but the VWP only allows those activities permitted on a Visitor Visa ((B) type):
An alien (other than one coming for the purpose of study or of performing skilled or unskilled labor or as a representative of foreign press, radio, film, or other foreign information media coming to engage in such vocation) having a residence in a foreign country which he has no intention of abandoning and who is visiting the United States temporarily for business or temporarily for pleasure.
The business part of which (B-1) does cover seeking to come to the United States for, amongst others:
Conference, meeting, trade show, or business event attendee
Will receive no salary or income from a U.S based company/entity.
For scientific, educational, professional or business purposes.
Lecturer or speaker
No salary or income from a U.S. based company/entity, other than expenses incidental to the visit. If honorarium will be received, activities can last no longer than nine days at any single institution or organization; payment must be offered by an institution or organization described in INA 212(q); honorarium is for services conducted for the benefit of the institution or entity; and visa applicant will not have accepted such payment or expenses from more than five institutions or organizations over the last six months.
Participating in a training program that is not designed primarily to provide employment.
Will receive no payment or income from a U.S. based company/entity, other than an expense allowance or expense reimbursement related to traveler’s stay.
So it seem quite clear that you are not in compliance except that another category is:
Independent research, no salary/income from a U.S. based source, or benefit to U.S. institution.
which I mention because it applied to me. A number of experts (in visa administration and immigration law) scurried about for a while to obtain a definitive answer. In my case that was that my activity counted as research and that I would be paid for it from a U.S.-based source made no difference. (An alternative source was readily available but was simply not required.) The subtleties may be very subtle indeed. Research was fine while there but not writing my report or, I think, even drafting it, while there.
I’d still interpret your situation as non-compliant. However, if that is so, you would not be eligible for a B-1 visa anyway (since the rules – whatever they may be! – apply equally). So if not eligible for a B-1 visa (eg arriving on a temporary basis to work, be employed, and/or be paid by a U.S. based company as a skilled or unskilled worker) you would need a temporary worker type of visa and the prospective employer must on your behalf file with U.S. Citizenship and Immigration Services (USCIS) a nonimmigrant visa petition accompanied by an approved labor certification.
Hence other than perhaps choosing to act as a whistle-blower it may be best for you merely to keep quiet.