That is not correct. North Carolina uses the same rule that many other jurisdictions do in counting calendar days. You ignore the first day (the day the judgment was rendered) but you don't otherwise exclude days, i.e. there is no excluding the last day, which would be a very odd rule indeed. Specifically, North Carolina GS 1A-1, Rule 6(a) states:
(a) Computation. - In computing any period of time prescribed or allowed by these rules, by order of court, or by any applicable statute, including rules, orders or statutes respecting publication of notices, the day of the act, event, default or publication after which the designated period of time begins to run is not to be included. The last day of the period so computed is to be included, unless it is a Saturday, Sunday or a legal holiday when the courthouse is closed for transactions, in which event the period runs until the end of the next day which is not a Saturday, Sunday, or a legal holiday when the courthouse is closed for transactions. When the period of time prescribed or allowed is less than seven days, intermediate Saturdays, Sundays, and holidays shall be excluded in the computation. A half holiday shall be considered as other days and not as a holiday.
(Bolding added.) You can see from what I put in bold the statute expressly states that last day is indeed counted. So if the judgment was rendered 8/3, the first of the 10 days starts on 8/4. If you do that counting the last day for filing the appeal is indeed 8/13, not 8/14 as you indicated.