I've built quite a few lambda + API gateway based apps lately, a couple are open source. Want to replace your DDNS service for your home network? Use this app, which will cost you $0 a month (not including the cost of a domain name): https://github.com/jmb12686/serverless-ddns
Developers call themselves “full stack” because they work on frontend and backend (let’s not debate how valid it is in their case) - in this case I think you’d be better off just referencing the specific tech your tool will support provisioning for.
Full stack is essentially like if a classic “lamp” setup tool just said “sets up your server”, and never specified what it sets it up for.
Interesting semantic discussion nevertheless.