This is really a terminological issue.
Is there a word for 3?
I am not sure if there is a word for no. 3, but I believe you are on to a genuine problem. My experience is that native speakers of any language “hear” what they think they ought to be hearing (in effect: a phoneme) and not what their ears are actually telling them. We had a discussion very recently on another site about the Arabic word phonologically /dʒanb/, phonetically [dʒamb], meaning “side”. A speaker on Forvo was definitely saying it as [djand], with [d]. The non-native speakers in the forum recognised this immediately, but the native Arabic speakers were insisting that they heard [dʒanb]. Have a look for yourself: http://forum.wordreference.com/showthread.php?t=2887668
What the hearer perceives must clearly be called a phoneme.
The produced phone will either be interpreted as the phoneme that the speaker intended, or it will be interpreted mistakenly to be another phoneme.
Typically, such mistakes may occur when a non-native hearer, or a hearer of a different dialect, interprets the phone as a phoneme of his first language and doesn't take into account a feature that the speaker takes into account.
For example, there's no /a/ phoneme in Finnish, it's either /æ/ or /ɑ/, but a French native speaker hearing Finnish words will interpret both [æ] and [ɑ] as his native phoneme /a/, and hear no difference unless he consciously pays attention to it.
This idea that the language influences what we hear has been called the "phonological sieve" or "phonological filter" (I don't know if there's a standard english translation) and has first been put forward by Trubetzkoy and what fdb talks about in his answer is the same phenomenon. Our brains get used to treating some phonemic features as relevant and ignoring others as irrelevant, depending on our first language.
Obviously there are limits to this. If the phone heard is phonetically too far removed from the prototypical sound of any phoneme of the hearer's language, it'll be interpreted as gibberish, or even as noise. That's how a European will hear a click.
This is a much more general problem of epistemology. Suppose a man strikes an anvil with a hammer and you hear it; then the epistemologists question would be, do you hear the man hitting the anvil, do you hear the force of impact between two hard surfaces, do you hear the acoustic waves coming from that contact? That debate has raged for centuries and is manifested in the splits between idealists and realists, and between direct realists vs. indirect realists. IMO the correct answer is "Yes, all of the above". So, you do hear the phoneme /s/.
Questions about phonemes are, of course, vastly less answerable since we can't even agree on the ontological status of the "phoneme". A "phoneme" is not necessarily an underlying segment (once a segment has been changed by some rule, it isn't what it was underlyingly), nor is it necessarily a segment that it to be changed into something else (so /t/ in "top" is eventually destined to be changed into aspirated t, but /t/ in "stop" is not ever changed to something else). Numerous theories of phonology distinguish between the phonological output and the articulatory implementation of that phonological output, in which case it would not be possible for a person to (directly) articulate a phonological segment -- that which is articulated is neither a segment nor phonological. (See Mark Hale Historical linguistics: Theory and method for discussion of the chain between speaker's stored representation and hearer's perception).
Since philosophers can't agree whether we can see a cow, rather than seeing light wave reflecting off of cow molecules, linguists can't be expected to have a clearer theory of what we perceive.
