First of all I'm new to Lattice and hasse diagrams theory.
I don't understand why this diagram is not a Lattice. Supposedly this is not a Lattice because not all elements have a join. Why can't $f$ be the join of $b$ and $d$ for example?
If we take a look at following diagram , we now have a Lattice. But the join for $z$ and $u$ is $x$. What's the difference?
Take $b, d$ for instance. $a\geq b,d$ and $e\geq b,d$, so if $\alpha$ is a lub of $b,d$ it must be smaller than both $a,e$. But the only elements smaller than $a,e$ are $b,d,c$. $b$ is not bigger than $d$, similarly $d$ is not bigger than $b$, and obviously $c$ is bigger than neither of them.
So $\alpha$ can't be a lub of $b,d$: since $\alpha$ was arbitrary, there is no lub of $b,d$: the displayed order is not a lattice.
In the second picture, the difference is that $x$ is the only element bigger than $z,u$ and so it is (almost vacuously) a lub of $z,u$: $y$ and $v$ are not as $a,e$ before.