Am I a British citizen overseas or just a UK British citizen?

I would guess that if you are not aware of the distinction, you are a British citizen. Which class of British nationality you hold would be indicated on your passport. For example, here is a British Citizen passport:

A British Subject passport:

And a British National (Overseas) passport:

Beside other differences, note that “British Citizen”, “British Subject”, etc. is explicitly written under the “Nationality” rubric on the identification page.

If it says anything else that “British Citizen”, you are not eligible for the visa waiver program and will need a visa to travel to the US. From the UK governement's foreign travel advice:

The US Visa Waiver Programme (VWP) allows most British Citizen passport holders to visit the US for up to 90 days. The types of journey that are permissible under the VWP include tourism, certain types of business visit and transit to another country. If you have a British passport that describes your nationality as something other than ‘British Citizen’ you will need to get a visa.

Beyond that, the requirements to get a visa will depend on the type of visa and while some applications might be assessed more carefully than others, formally I don't think they differ depending on citizenship.