In the old days there used to be 29 letters.
The "alef" and the "hamza" are separate.

• "Alef" used to refer to the silent vertical line letter in long fathas. It can't take ANY haraka on its own (much like the silent waw/ya can't, in long dama/kasra). It's not even an independent letter - to draw it they associate it with another letter, often lam, this way لا. Interestingly it was dead last in old alphabet order, with hamza first.
• "Hamza" used to refer to the /a/ letter which could take harakat.

Hamza has many types: * Hamzat Wasl (linking hamza) - like the one in your example. As I said, since it's a hamza it can take harakat (that's the difference between it and alef, otherwise they're identical in shape). However if you noticed, it's a vertical line which lacks that "number 2"-shaped piece (which is accordingly called "kit3a", piece in arabic).
That's how we know it's a hamzat wasl, alongside the fact it's not pronounced when in the middle of sentences (instead morphed and elided with adjacent letters).
Its more well known use would be the "al-" article (often it's drawn in this case with a "sad letter-like" tiny shape above it, it's in fact a fancy-shaped fatha) and imperative verbs, among other things. * Hamzat Kat3 (Cutting Hamza) - 90% of other hamza types. Depending on what the "kit3a" sits on, it has different names: hamza ala alif (on alif), hamza ala waw, hamza ala nabra (above dot-less ya mid-sentence), and hamza ala satr (on the line, alone). It's the same letter, you choose one of these forms following one simple paper-rock-scissors game involving the haraka on the hamza and the haraka on the letter before it (if any).

One letter word hamza with fatha is quite often used as a calling tool ("Hey you, ...!") and an interrogation tool ("Is it true that..?")), and might be THE iconic and most natural written representation for the hamza is written...
* It's not an imperative verb, nor "al-", nor one of these other weird words.
So... Hamzat Kat3. * Kasra (nabra) is stronger than dama (waw), which is stronger than fatha (alef), which is stronger than sukun/nothing.
Our hamza has a fatha. Previous letter (nothing) has nothing.
Fatha is stronger than nothing. Alef wins.
So... Hamza ala alif.

Meaning hamza alone has a natural penchant for being drawn as the "kit3a" (piece) sitting on the alef (the vertical thing). Linguistics noticed this and then started saying weird things...
Like calling alef as "alif layina" (soft alif)
and calling hamza as (gasp!) "alif yabisa" (rigid alif)

Eventually, people just settled on 28 letters, and calling both alif and hamza with the "alif" name. Calling the hamza (no matter which type) as "alif" is technically an abuse of language, but is widely used and accepted even among linguistics.

The reason I typed this long-ass post was that I'm seeing the old and new terminolgies being lumped against each other and this makes the deal much more confusing than it needs to be.