That is a legitimate criticism, you are essentially proposing that we treat religious adherents in a similar way to the way we treat countries, if the relevant actors consider you X then you are X. Your definition has problems too though.
First by your definition there is evidence of Catholics (Catholic priests no less), who are atheists. If a Catholic priest loses their job often they don't have much to fall back on, and apostasized priests have kept preaching believing that the little lies they percieve themselves as telling are bringng people hope. While they might have moments of candor with those close to them their public face is one of a good Catholic priest who most other folks (including the Supreme Pontiff) would recognise as Christian.
Next you have the problem of what constitutes 'mainstream'. The early Jewish sect that sprung up after Jesus had comparatively little organisation and structure (which was far more congruent with Jesus' teachings). But a few centuries after Jesus death it had all but vanished. By your definition the religious movement which more closely matched the religion of Jesus' would not be Christian as by that point the Roman Empire had spread Paulian Christianity to much of Europe, North Africa and the Middle East.
A final problem, and this is more technical, is that typically definitions of descriptive nouns / adjectives like this refer to properties of the thing they describe, not properties of someone else describing the thing. A ball is considered a ball because it matches the conditions to be a ball (being sufficiently round, being sufficiently small, etc). The reason for this is that ultimately the meaning of words is derived from their usage (to convey meaning some set of people have to use a word), so it is basically tautological to imply someone can be described as 'a noun' or a 'adjective person' if the right people use the word that way as what we are asking isn't the fundamental reason words have meaning. It is like answering the question of why my toast popped out the toaster early this morning with 'the big bang'. Sure technically true, but completely irrelevant. In this instance you are arguing for specific authorities to determine the meaning of words which is a common way to circumvent this problem, but that just brings us back to the last paragraph. I'm out in cricket if the umpire says I'm out. Here the authority is not so clear. Are Southern Baptists Christians? What about Christian heresys that explicitly reject central authority like Catharism, were they Christian?
You do have a point though, I don't consider someone a Christian just because they claim to be one, I don't think it is enough to just claim you are a Christian I would say you are a Christian if you genuinely believe you are following the teachings of Jesus. Now this has some interesting consequences. For instance under this definition I don't think most of the Westboro Baptist Church are Christians. If you look at how they operate they are clearly a legal scam who likely don't believe a word they are saying. Enough members of the KKK and DAESH are Christian and Muslim respectively under this definition (which coupled with the groups making religion part of their mission statement), so I will call them Christian and Muslim.
But ultimately however we define words like 'Christian' and 'Muslim' we wont find a perfect definition. So if these are problems you can live with and you like your definition that is fine. I admire what you are trying to achieve with your definition, to make it clear to people that adherents of Islam are not fundamentally distinct from people in the West, and that extremists like this don't represent your typical Muslim. You want to avoid labelling the extremists of a particular religion by that religion because it has the odd property of putting me in the same box as Stalin, the nice lady at the soup kitchen in the same box as Raynald of Châtillon and al-Khwarizmi in the same box as Osama Bin Laden.
You are right, as far as I know DAESH aren't supported by any mainstream Clerics. I wish the media would report on them in a similar way to the ways the report on the KKK, bringing up their religion only when it is relevant, I think some reporting at the moment is irresponsible ignoring that most of DAESH victims are Shia Muslims with no interest in this so-called 'Jihad'. But I'm unwilling to use language in such a way that we completely ignore the religious motivations of the participants because like the Crusaders, like the KKK, like many religious extremists, religion is playing a role in motivating DAESHes actions.