The terms "web developer" and "software engineer" are a bit muddied and it's not really anybody's fault (just years of inadvertent misuse and abuse as non-industry folk such as recruitment agencies have struggled to describe what it is we do).
There are different types of software engineer:
And there are also different types of web developer:
So for example, let's take Twitter. Twitter has both engineers and web developers. The engineers are busy writing the backend and exposing its APIs, while the web developers are creating its front end (not just client side!).
In contrast, let's take a sole PHP developer working on a bespoke product. He/she is both an engineer and a web developer, creating both the backend software and the front end.
In contrast again, let's take a sole PHP developer creating a website for a client by customising Wordpress (or Shopify, or something off the shelf). Are they an engineer? If they are not actually architecting either the back or front end, then I would be tempted to say not.
So, to cut a long explanation short, these days you have to take the context into account to determine what they are really looking for. Not all engineers are web developers (obviously), but also not all web developers are engineers, and the distinction really lies in the role itself and its requirements.
In your example above, I would say that Google's interpretation of and distinction between "web developer" and "software engineer" are pretty clear cut as they make those distinctions in the projects that they publish and contribute to (Angular, Kubernetes, etc). A "web developer" at Google will almost certainly be working primarily on the client side, or on the client side of the server side, whereas a "software engineer"'s jurisdiction will almost certainly end at the HTTP API.
I hope that helps, though I realise it doesn't clear much up - it's a bit of a problem in the industry really, and has been for a while, but we seem to be muddling through ok-ish...