In general, the name as first officially recorded at birth has the most influence. "Officially" being a religious, governmental, or familial book or register where babies' new names are recorded.

The numerological influence of the second name depends a great deal on how much the person identifies with it.

The amount of influence doesn't depend very much on how other people identify the person as, nor how long the person has had the name. Instead, it depends most on how much the person identifies themself as that name.

With a birth name and a common name, both are influential. It's a blend.

A consideration is how compatible the two names are with each other. Get the core numerology numbers of the two names and compare the numbers for compatibility. In general, the more compatible the two names, the less personal struggle with ideals, goals, and perception of purpose.

Other numerology chart positions have an influence on the individual, such as the birth date number mentioned at the core numbers article, linked above. More on the influence of the birth date can be found in the Numerology and Your Life Path article.