Which definition of Acids and Bases conforms to the pH scale?
A few things first...
Arrhenius Theory:

An &quot;acid&quot; is a proton donor; a &quot;base&quot; is a hydroxide donor.

So for example, $\ce{HCl}$ and $\ce{NaOH}$ (both in water) ionize to produce $\ce{H+}$ and $\ce{OH-}$ respectively.
$$\ce{HCl -&gt; H^+ + Cl^-}$$
$$\ce{ NaOH -&gt; Na^+ + OH^-}$$
Which gives us the all-too-familiar neutralization reaction:
$$\ce{HCl + NaOH -&gt; NaCl + H2O}$$
Brønsted–Lowry Theory:

An &quot;acid&quot; is a proton donor; a &quot;base&quot; is a proton acceptor.

At first, the base being a &quot;proton acceptor&quot;, may appear to be in contradiction with the Arrhenius Theory... but it isn't.

 The following's from Chemguide. It's been put so well, that I didn't even bother with paraphrasing it 
The Brønsted–Lowry theory doesn't go against the Arrhenius theory in any way - it just adds to it.
Hydroxide ions are still bases because they accept hydrogen ions from acids and form water.
An acid produces hydrogen ions in solution because it reacts with the water molecules by giving a proton to them.

When an acid in solution reacts with a base, what is actually functioning as the acid is the hydronium ion. For example, a proton is transferred from a hydronium ion to a hydroxide ion to make water.
$$\ce{H3O^+ + OH^- -&gt; 2H2O}$$

 I'm assuming you already know what the hydronium ion is...so I won't get into that 

Or, meh, I'll re-phrase that last bit. Remember that example/equation we first  brought up?
$$\ce{HCl + NaOH -&gt; NaCl + H2O}$$
Looking at it from the Brønsted–Lowry POV, your &quot;acid&quot; is really just the hydronium ion, because that's the real proton-donor here; the &quot;base&quot; is really the hydroxide ion, because that's the real proton-acceptor here.
The hydronium ion (acid) passes on a proton to the hydroxide ion (base).
$$\ce{H3O^+ + OH^- -&gt; 2H2O}$$

Back to your question:

Which definition of Acids and Bases conforms to the pH scale?

Both the Arrhenius and Brønsted–Lowry definitions are &quot;compatible&quot; with $\mathrm{pH}$ . They both point in the same direction ;)

Our teacher told us that pH stands for power of Hydrogen, but some sources say it is a number. Which is it?

$\mathrm{pH}$ is, by definition, (approximately) the negative logarithm (base 10) of the molar concentration of $\ce{H+}$ ions in a solution.
Therefore, $\mathrm{pH}$, is indicative of the $\ce{H+}$ strength, in a solution.
I've been led to believe that the $\mathrm{p}$ in $\mathrm{pH}$, comes from the German &quot;Potenz&quot;, or the French &quot;Puissance&quot;, both which mean the English &quot;Power&quot;. I'm under the impression it was originally &quot;Potenz&quot; since German was the language of science well up to the end of the 19th century (and then the whole World War thing ruined that...); besides, the man who came up with the $\mathrm{pH}$ notation was a Dane, and Danish is a Germanic language with little semblance to French. However, Wikipedia suggests it's more likely &quot;Puissance&quot;, since the laboratory the man worked for was French-speaking ...that's still just a guess though.
Your teacher isn't wrong, the $\mathrm{p}$ does mean &quot;power&quot; (or even &quot;potential&quot;).
However, $\mathrm{pH}$ is a number (it is dimensionless/unit-less...but not without meaning either), so your &quot;sources&quot; are correct too ;)