Can a source be considered 'primary' if it is not written by an eyewitness and partly based on another source?

A "primary" source is the first or original source of information about a fact. It has nothing to do with time. Even if an event occurred hundreds of years before, the source is still primary if it is the first or original source of information.

For example, let's imagine Henry Neville (1564 – 1615) wrote a letter in 1594 which said that George Neville (1440 – 1492), one of Henry's ancestors, killed John Beufort at the Battle of Tewksbury (1471). This letter would be a primary source for this fact even though the event occurred over 120 years before the letter was written. If a historian later wrote a book about the War of the Roses and included this fact, citing the letter as evidence, then the book would be a secondary source for the fact. Secondary sources cite primary sources as evidence. Primary sources do not cite any prior sources because they are the first source of information.

Since the New Testament, including the Gospel of Luke, records many facts for the first time, it is a primary source for those facts. In fact, the very name of the Gospels, the New TESTAMENT, indicates the role of the Gospels as a primary source. A testament is the account of a witness. Any account of a witness is a primary source, even if it is written down long after the event being witnessed occurred.

Additional Commentary

(I am writing this additional commentary because some people have been relunctant to accept my apparently novel definition of a primary source.)

First of all, there are very few books on historiography, and those I have read I have found to be universally ill-conceived and illogical. Many of them do not even have glossaries or any definitions of key terms at all. Most books on historiography currently in print have no definition of a primary source at all, and those that do define it only in an ill-considered offhand way as a "eyewitness testimony" or "a contemporary record" both of which are completely incorrect. Many just list "types" of primary sources (newspapers etc) without even defining the term.

Let's take a newspaper, for example. Is that an eyewitness testimony? No. Newspaper reporters arrive AFTER the fact, then question (supposed) eyewitnesses who then tell them all sorts of lies, half-truths and misapprehensions which the reporter then writes down. I have been interviewed many times by newspaper reporters and NEVER even once have these reports either reported my words verbatim or even according to their meaning. Nevertheless, newpaper accounts are considered "primary sources."

Furthermore, most sources of history are written long after the events in question. For example, the Saxon Chronicle, the most single important PRIMARY source of British history is the source for large numbers of events which occurred hundreds of years before the Chronicle was written. The entire fabric of history is written from such sources.

The best definition I have found published for a primary source is that of Aims Community College in Colorado:

A primary source is an original study, document, object, or eyewitness account. In other words, this is the source where any given information first appeared. For instance, if a scientific study is performed, the primary source is the initial report that is prepared by the scientist(s) who performed the research.

It is kind of a joke that a community college got the definition correct, but if you look at handbooks published by Harvard, Yale and Princeton they make incorrect definitions. Maybe some of their professors should take some courses at Aims and learn something.

The underlying problem here is that historians have a very haphazard and unanalytical approach to studying history (compared to intelligence analysts, e.g.) and consequently their own definitions and understanding of information analysis is lacking, something pointed out by von Ranke over 150 years ago, and Gibbon before that. A book needs to be written on information analysis that clearly spells out how to systematically analyze and assimilate information. Unfortunately, that book is not yet written.