Are cell signal boosters & wifi signal boosters considered routers?

Not quite. A wifi signal booster is more similar to a wireless networking router than a cell phone signal booster is. A cell phone signal booster will not help a broadband WIFI service (DSL nor Cable) as they are designed to work with mobile phones and devices. WIFI in the home uses a different frequency and needs a Wifi booster to be improved. Let us explain in detail below.

A wireless router is a networking device that forwards data packets between computer networks. For example, it received Internet data from Internet Service Provider (ISP) and redirects it to various areas of the home where a networking cable can be connected to a computer in a specific room where Internet can be accessed.

A cell phone signal booster sends wireless signals from outside the home or building to an amplifier inside which then amplifies that signal and broadcasts it inside home to different areas through interior antenna(s). Therefore, a cell phone signal booster basically amplifies wireless signals distributes them indoors.

A wifi signal booster on other hand is a bit more similar to the wireless router in your home than a cell phone signal booster. In fact, a wifi signal booster is like having two wireless routers because one of these wireless routers picks up existing WiFi network (data packets being released wirelessly from your wireless networking router). Then it redirects those wireless data packets to the other wireless router, which transmits the wireless data packets expanding the wifi coverage area.