Depending on how one counts, Spanish has a great deal more than three forms for commands.
There are only three true imperative forms of each verb (I'll use the verbs hablar, comer, and vivir for my examples throughout this answer):
- tú: habla, come, vive
(tú form less the -s; several irregulars: di, haz, pon, sal, sé, ten, ve, ven; estar always reflexive: estate)
- vos: hablá, comé, viví
(vos form less the -s; one irregular: ir is andá)
- vosotros: hablad, comed, vivid
(infinitive, less the -r, plus -d if non-reflexive; one irregular: irse is idos)
In older or very formal Spanish, the vos commands will be identical to the vosotros commands (they were derived from that form) and may retain the -d- in reflexive commands.
Before I get started on the rest, I'd just like to say don't oversimplify English commands. They are actually a good bit more complicated than just adding "use the bare infinitive for affirmative, add don't to create a negative." Many can take the word not after them to create the negative sans do ("fear not"); the first-person plural commands are formed with let's which creates an embedded clause ("let's play"); and when using don't or do not (for emphasis), you are actually switching to the emphatic mood which is also often used in affirmative commands ("do mind your manners"). And we don't even really have a way to give commands to people we aren't talking directly to, except by using a variety of structures using the modal verbs let ("let there be light") or may ("may they live happily ever after") which could be seen as either an impersonal command (let) or a fossilized subjunctive (may). In any case, English commands are actually really quite complex when you think about it.
Back to Spanish. Spanish could easily do third-person commands by using the subjunctive (nowadays, it's most common though not required to preface these with que; when used sans que it's called the exhortative subjunctive). Usted back in the day wasn't a true pronoun, just a third-person honorific, so it was treated as any other third-person word.
For negative forms, Latin — out of which Spanish developed — didn't have a negative imperative: like English, it used a separate verb's command to form negative commands (noli(te) — from nolo — meaning "want not"). Spanish innovated a different form, using the subjunctive (perhaps derived from (no quiero que) [verb] or similar phrases). This ended up being applied to all commands, first, second, or third person.
subject affirmative negative (prefix all with "no")
tú | habla come vive | *hables *comas *vivas
vos | hablá comé viví | †*hablés †*comás †*vivás
usted | *hable *coma *viva | *hable *coma *viva
nosotros | *hablemos *comamos *vivamos | *hablemos *comamos *vivamos
vosotros | hablad comed vivid | *habléis *comáis *viváis
ustedes | *hablen *coman *vivan | *hablen *coman *vivan
* derived from subjunctive
† in Chile: *habléis, *comáis, *viváis with an optionally elided -s.
As to why the subjunctive form appears to be switched or is based off of the yo form verb, the reason is simply because that's how it was in Latin. As to why Latin did it, generally when asking why in linguistics the answer is simply "because that's what people said".