DC's Legends of Tomorrow (TV Series)

So, I'm not sure if this question has been asked already but I've just finished watching Legends of Tomorrow S07E10, and Eobard Thawne shows up there.

SPOILERS!?!?!?!?!?!?!?

In that episode the Legends are trying to prevent WWI by preventing the assassination of some Archbishop dude, and it turns out that that's impossible: it's a fixed point in time. Anyone who tries to alter a fixed point is unable to, and is usually met with unfortunate consequences. Sara Lance is who has been selected (or she nominated herself because she can heal from anything) to prevent this assassination. She tries innumerable times and although she gets close, she fails. It turns out what's been preventing her (and everyone else) from saving this Archbishop is Eobard Thawne.

Eobard explains that after being defeated by the Black Flash, back in S02, the Time Wreaths had had other plans for him: appointing him as the one who would make sure certain fixed points in time (the aforementioned assassination) are never altered under any circumstances.

My problem, now, stems from the above explanation because, if The Flash and Legends of Tomorrow take place in the same universe (pre and post Crisis), why is it that Eobard gives that explanations to his existence, and doesn't refer to The Flash's S05 and S07's appearances? Who's the character that shows up in The Flash then (who later shows up closer to the end of The Flash S08)?