There is no right or wrong answer for this question. It depends on many factors, including the plans you have for the size, scale, and geography of your business.
Having the word in your business name will immediately identify your business as working in this particular niche. But that is potentially only useful if customers have some prior understanding of the term.
It is highly unlikely that someone who does not recognise the word would be sufficiently stimulated or intrigued upon seeing your business name containing "detailing" to initiate contact to find out what you do. You would rely on other parts of your brand to do that. Perhaps a strap-line. Perhaps your website content. etc.
If you plan to get business via word of mouth, you can argue it's entirely irrelevant in the title. If you intend to be found via your website & search engines, utilisation of it as a keyword is really important, and so including it in your title could be important, but having good references to detailing in the website content is probably just as valuable. I say "could" above, as it is highly unlikely that your business would get ranked higher than other sites just because of the word detailing in your business title alone. I'm pretty sure your business would not fail just because you neglected to include "detailing" in the name.
Personally, I wouldn't sweat too much on trying to assess the technicalities of this. I'd spend some time coming up with a brand name I was really comfortable with, and if that so happens to end up with the "d" word in there, then so be it! I like the word, so if I ever set things up I"m pretty sure it would be "f4780y detailing" - now there's a can of worms
Good luck with the business!