The terminology “pointed $(\infty,1)$-category”, in this sense, is commonly used, for instance, when speaking about stable (∞,1)-categories, which are such pointed $(\infty,1)$-categories with further properties.

If a pointed $(\infty,1)$-category in this sense happens to be just a 1-category, then it is a pointed category.

The same terminological caveat applies as applies to “pointed categories”:

More generally, a pointed (∞,1)-category could be taken to be a pointed object in (∞,1)Categories, i.e. an (∞,1)-category with any of its objects singled out, and with (∞,1)-functors between such pointed $(\infty,1)$-categories required to preserved these chosen objects.