The simple fact of the computer's presence in the present is itself the source of endless fascination, but not where I wish to go right now. Gideon highlights a specific point which I feel will require some thinking through to reach a generally acceptable compromise for users. It is a question of timelines, and when to refer to the single individual identified as the Reverse-Flash as Harrison Wells, and when it is more appropriate to label him Eobard Thawne.
Gideon herself addresses this entity as "Dr. Wells" within the context of the present and his 'secret chamber', although she must be aware of his 'real identity' (one assumes, unless the computer's memory was somehow tampered with) and there is no one to overhear. Likewise, through the first half of the season and more, all the characters have believed him to be Harrison Wells, whether genius philanthropist or enigmatic potential killer, so it feels ... uncomfortable to me to rewrite that particular form of history to permanently label him "Eobard Thawne".
What I'm floundering around in the vicinity of, I think, is that we may need to agree some (slightly arbitrary) point as the 'dividing line' between the two, and then standardise references on each side of that probable chasm. Someone else may have a better solution, or at least so I hope, but I believe it's a point worth discussing.