Starting with the Black to Transparent as you have in the editor, click once above the gradient (my green highlight).Then drag the Opacity Midpoint slider to the right to soften the transition and move it to your required location.To save the new gradient give it a name (where it says Custom) and click on New.

You can create a foreground to transparent gradient. https://imgur.com/a/oxOX4MA

Wrong way. It seems you are trying to use Window > Gradients panel. Try to activate Gradient tool as already explained in reply above, then choose gradient from the Options bar and lastly drag gradient on canvas while mask is selected in the Layers panel. You can either use transparent/black or white or black to white gradient.

Here's how I would achieve that effect:
1) Create a BW Effects layer, and select the mode that gives the most pleasing effect.
2) Edit>Copy Merged, then paste that into a new alpha channel
3) Invert the alpha channel
4) Delete or disable the bw effects layer
5) Select the active pixels (Control key (PC) or the Command key (Mac) while you single click on the alpha channel in the channels panel)
6) Create a new solid color layer using the purple you used above

The active selection of the alpha channel will automatically be applied as a transparency mask. That SHOULD do it. Good luck.

Even though you have applied a black to transparent gradient, via the layer effects, to your top layer, I can see that the layer itself is still filled with white.

Set the fill of the top layer to 0%

Another tip for you: Change both colors of your gradient to black, then adjust the opacity. Leaving the gradient as a Black > White one will mean that the colors between the two points are not only black, but varying shades from black to white and therefore not a true black to transparent gradient

Here you can see what you have right now: a white layer with a black > transparent gradient style applied to it. As you can see, the fill color of the layer is still white

You need to decrease the fill of the layer to 0%, which will make the layer transparent, allowing your transparent gradient to work

A simple break-down:

• Fill is the base upon which your layer styles are applied. No matter what styles or effects you add, the fill is still underneath it all. Change this to affect only the fill of the layer and leave your styles intact
• Opacity affects the entire layer (fill and styles). Change this to decrease the opacity of everything on the layer

Use the image as an alpha mask. Take this black and white "logo" as an example:

1. Select All and Copy (or Copy Merged if you have multiple layers).

2. Create a new mask with the "New Layer Mask" button in the Layers panel (either on the your logo layer or a group containing your logo if it has multiple layers).

3. Alt+Click the mask thumbnail to edit the mask (your canvas should turn completely white), then paste the image you previously copied. Click the layers thumbnail to finish editing the mask.

4. As you can see, the semi-transparency defined by the various levels of gray are still gray in color too. If that isn't what you want then select all and fill the layer completely with white (or whatever color you want the logo to be).

Your now transparent logo:

If it’s a Gradient Fill layer, when that layer is selected it should be possible to adjust its properties using the Properties panel, as you see in the picture below. In this way you don’t have to open a stack of dialog boxes; that’s why they recently made this improvement to make the options of more layer types immediately accessible in the Properties panel. In the picture below, you can see that I added a middle opacity stop set to 0% opacity. One thing that can confuse people is that the option they want, such as Opacity Controls, might be available in the Properties panel, but not visible. When this happens, make sure the options group you want is both scrolled into view (in case it’s hidden further down), and if necessary, also expanded (if it’s collapsed). That’s the reason my screen shot is so tall. If it was shorter, Opacity Options would have been covered by the Layers panel unless I scrolled the Properties panel to bring the Opacity Options into view.  If the gradient was applied another way, it might be different. For example, you’d edit a Gradient Overlay effect in the Layer Style dialog box.