We need to help you clarify a couple of things first.
The focus keyword is just the only way Yoast SEO have to know about what is your targeted keyword for that particular page or posts. If you do not define what your focus keyword is, Yoast SEO will not know what is the keyword you want to be focused on that particular content and therefore will not know how many times that focused keyword appears in the content or the meta title or meta description tags.
Yoast SEO will assign and score or green color if everything is “optimised” including the number of times you focus keywords appears. They said that it will overwrite the meta keywords because the will use that instead of the keywords you use in the meta keywords.
Seobook web crawler won’t have access to the focus keyword because this is for Yoast SEO only to know. To put it in a way, The focus keyword is for internal use. That’s the reason why they only return the meta keyword tag which, by the way, Yoast allow you to also control and by default is public.
Yoast SEO allows you to setup also the meta keywords tag, as much as allow you to setup the meta title and the meta description tags. All these three tags are not for internal use, is for everyone else to know and see.
In other words, the focus keyword is just a tool that Yoast provide to help you count the number of times your keyword is visible in your content. It will make more sense if you go and try.
The meta keywords tag is public, and people do not use it anymore for SEO, it’s obsolete and if you list all the keywords that you will like to rank for in there everyone who uses the seobook web crawler will find out which keywords are you trying to rank for, competitors? Maybe?
I recommend you not to use it, just focus in writing good content around a topic, that’s much better for SEO and don’t make grammar mistakes like I probably did here ;)