You should choose Corel draw or illustrator to design logos for your websites, they will generate perfect images in vector graphic which Photoshop couldn't do, it's for bitmap vector and common graphics
Use any tool, mostly Photoshop is design friendly tool.
Logo designing is a creative, hardworking and professional task.
In real world scenario, we have to offer three kind of options to our clients.
Only text, text with icon or only icon.
In text related logos we can use awesome fonts according to their logo or brand name.
Even in icon matter also we have prepare a hand painted design, then only we can go for design.
Designing logo is a vector-based work, so we'll have to use Illustrator or Corel Draw. For me, Illustrator is better, cause it linking well with other essential Adobe tools like Photoshop, After Effect, Adobe Audition.
All Adobe Products you've mentioned are standard softwares for most professional graphic designers.
If you want to create a logo painlessly, I recommend Canva. It has everything at your fingertips. With the Adobe products, you're gonna need a library of images, etc. to start with... which is time consuming.