GSM modem = USB GSM modem to use with carrier´s sim card. Can use one or more.
GSM Gateway = appliance that provides higher density of sim cards, providing sometimes network interfaces, support to http api or even SMPP protocol. E.g. GoIP, Dinstar, etc
"external carrier" = SMS aggregators that provides HTTP API or SMPP protocol acces. E.g. Infobip, Nexmo, Clickatel, etc.