I don't think any linguists would define vowels as strictly being produced with absolutely no obstruction whatsoever. If any do, it's an exaggeration. There aren't any such clear cut distinctions - there's a gradual opening of the mouth from stops through fricatives, approximants, and vowels. This is what is called the Sonority hierarchy. Wikipedia gives a simple table there showing how various terms relate to each other. One thing it points out is that vowels are syllabic: they form the centre of syllables. But this isn't a hard distinction. While most consonants are non syllabic, in some languages some consonants are analysed as being syllabic, including in English.
The mouth does have to actually produce different sounds, so anything other than the most open vowel ([a]) has to be less open because of the physical limitations. What distinguishes the vowels from the approximants is largely a matter of convention. We consider both how sonorant and turbulent the sound is, as well as the phonotactics of each language. The cross-language generalisations we produce are only that, generalisations, not hard rules.
One last thought: some languages probably have situations in which allophones cross the divide between vowels and consonants (though I can't remember of any examples.) Analysis of allophones and which is considered the 'default' is tricky at the best of times, so it's just another thing which you can expect to see in individual language's phonologies. But these edge cases don't mean that it's not useful to still have the basic categories of vowel and consonant.
