The correct interpretion of R is on a population-based, statistical level. In your example, if two infected people have contact with one person who becomes infected, we do not say they each have an "individual R" of 1, we say the population has an R of 0.5: 2 infected people result in 1 infected person.
Because you are basing R on population level statistics, you would never count an infected person multiple times based on multiple contacts, they are just one infection.
Sometimes R might be estimated from contact tracing, and in this case the situation you describe may be a problem, if you assume everyone in contact with a patient who goes on to get infected was infected by that person. Breban et al addresses this concern and is strongly relevant to your question (there might be better resources, too, this was just the first I found). The contact tracing method is more accurate if you have very few infected people such that it is unlikely that someone would be exposed to more than one possible source.
Breban, R., Vardavas, R., & Blower, S. (2007). Theory versus data: how to calculate R 0?. PLoS One, 2(3), e282.