What is the equation of acid base reactions?

I recognize:

HA + H₂O <---> A⁻ + H₃O⁺

Is this the correct equation?

Because sometimes I see

HA + H₃O⁺ <---> A⁻ + H₂O

Are they two different equations?
Update: Oh dayum! I've been writing doing a lot of nothing then!

So if I add 1.0M of HCl, it's the same as adding 1.0M H₃O⁺ and also 1.0M H⁺.
Update 2: Oh the other equation is HA + OH --> A + H3O
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