@fisicx
'Design' or 'designer' is generic... and, so what.... it's also a very well known term. Every person on here who specialises in a form of design will refer to themselves as a designer, either with an additional part such as graphic or web at some point in their career.
From a very simplified standpoint a 'designer' can do everything you listed, can they do it to the level of someone who has more specialist training or experience, no, but they can do it and depending on the requirements of the client they may be able to do the work they're being asked to do. I can tell you first hand (my background is product design) that if you ignore the engineering/manufacturing side of the process any capable 'designer' can turn their hand to designing furniture, clothes, cars or any other thing that requires a person to be creative.
I also think you're starting to get confused with the term tagline, you don't need a tagline on or with a logo, especially if you're focusing on the website like you seem to be. Including 'design' in the logo is not a tagline, a tagline is something that goes with the logo, think 'taste the feeling' that coca cola uses for example.
Besides you have so many other ways to draw attention to the services you offer on a website, designers have 'artwork' they can show, you can use good seo orientated written content and even the design of your site itself to draw attention (imagery, bold text, different size text etc) to the major areas of your expertise. At the same time this would help with search results, even more so if you have a dedicated 'services' page. You know those things YOU have mentioned to other people wanting critiques on their websites and then got reminded about when you did your own template....