The "I know it when I see it" features of bamboo are the persistent woody culms with side-branches (potentially) at each node.

The family Poaceae are the (true) grasses. This family is divided into 7 sub-families. One of those sub-families is the Bambusoideae, which is divided into two groups, group Bambusodae, and a group of other grasses called Oryzodae, which is divided into five tribes and includes some of the so-called wild rices. Bambusodae is divided into three tribes: Bambuseae, the woody bamboos, comprising 9 sub-tribes, and two other small tribes. This complicated taxonomy tends to suggest that bamboo as popularly understood may turn out not to be a valid taxonomical category in the sense of evolutionary biology, like whales and dolphins, or birds and reptiles, or mushrooms and toadstools, or butterflies and moths.

By saying "Arundo is in the same family with bamboo" you are saying is "Arundo is a true grass", which is correct, but no more than that. The grasses in my lawn are also in the same family with bamboo.

There are grass-like plants which are not Poaceae, but in the wider plant Order Poales. Poales includes some things which might colloquially be called grass, such as sedges (Cyperaceae), rushes (Juncaceae) and restios, but also a few things less likely to be mistaken for grass, most significantly the bromeliads and pipeworts. Perhaps this was the distinction you had in mind, ie, distinguishing Arundo from rushes and sedges. "Reed" is not a biological category, it is just a name applied to certain plants.