As pointed out by the other answers, every smooth weak Fano is certainly log Fano. But you also asked for an example of a log Fano variety that is not weakly Fano.

Take any toric variety for which $-K_X$ is not nef.

More explicitly, rational ruled surfaces (ruled surfaces over $\mathbb{P}^1$) can have $-K_X$ big but not necessarily nef. For instance, if you are doing the rational ruled surface with respect to $O_{\mathbb{P}^1} \otimes O_{\mathbb{P}^1}(e)$ for $e \gg 0$ (I think any $e > 2$ works actually). Check out the section in Hartshorne (or Lazarsfeld's Positivity book if you need background on ruled surface). It's a fun exercise to explicitly work out what the $\Delta$ divisor is (can be) that makes $(X, \Delta)$ log Fano.

As pointed out by the other answers, every smooth weak Fano is certainly log Fano. But you also asked for an example of a log Fano variety that is not weakly Fano.

Take any toric variety for which $-K_X$ is not nef.

More explicitly, rational ruled surfaces (ruled surfaces over $\mathbb{P}^1$) can have $-K_X$ big but not necessarily nef. For instance, consider the rational ruled surface with respect to $O_{\mathbb{P}^1} \otimes O_{\mathbb{P}^1}(e)$ for $e \gg 0$, this will not be weakly Fano (I think any $e > 2$ works actually). Check out the section in Hartshorne (or Lazarsfeld's Positivity book if you need background on ruled surfaces).

It's a fun exercise to explicitly work out what the $\Delta$ divisor is (can be) that makes $(X, \Delta)$ log Fano in the ruled surface case. You can also see explicitly why ruled surfaces over curves that aren't $\mathbb{P}^1$ cannot be log Fano.

As pointed out by the other answers, every smooth weak Fano is certainly log Fano. But you also asked for an example of a log Fano variety that is not weakly Fano.

Take any toric variety for which $-K_X$ is not nef.

More explicitly, rational ruled surfaces (ruled surfaces over $\mathbb{P}^1$) can have $-K_X$ big but not necessarily nef. For instance, if you are doing the rational ruled surface with respect to $O_{\mathbb{P}^1} \otimes O_{\mathbb{P}^1}(e)$ for $e \gg 0$ (I think any $e > 2$ works actually). Check out the section in Hartshorne (or Lazarsfeld's Positivity book if you need background on ruled surface). It's a fun exercise to explicitly work out what the $\Delta$ divisor is.

As pointed out by the other answers, every smooth weak Fano is certainly log Fano. But you also asked for an example of a log Fano variety that is not weakly Fano.

Take any toric variety for which $-K_X$ is not nef.

More explicitly, rational ruled surfaces (ruled surfaces over $\mathbb{P}^1$) can have $-K_X$ big but not necessarily nef. For instance, if you are doing the rational ruled surface with respect to $O_{\mathbb{P}^1} \otimes O_{\mathbb{P}^1}(e)$ for $e \gg 0$ (I think any $e > 2$ works actually). Check out the section in Hartshorne (or Lazarsfeld's Positivity book if you need background on ruled surface). It's a fun exercise to explicitly work out what the $\Delta$ divisor is (can be) that makes $(X, \Delta)$ log Fano.