Should my link shortener go on a single domain?

I'm building a very niche link shortener. The objectives of the project are 3 fold.

1. A small project to get me coding and launch something.
2. This link shortening will be a feature in my bigger long term product.
3. I'm giving it away for free, building my email list and if people shorten links with it, I will have a bunch of links coming back to my domain. If potential users are curious, they might investigate the root domain and signup themselves.

I currently run a site called at http://atmssg.com which shares content so digital marketers can learn how to use messaging channels for marketing.

The link shortener allows people to build deep links into SMS conversations. The target market is people that run SMS marketing campaigns. (links don't have to be short btw)

I was thinking about using a different domain for the link shortener, but just attended a really good pitch about digital marketing and one important point was that everything should go on a single domain.

So my plan is:
atmssg.com - is the blog and content site where I am driving newsletter signups
m.atmssg.com (or maybe a pun like f.atmssg.com) is the application homepage and the web app itself.
Links will shorten to m.atmssg.com/l/slug - and these are the links that people will create and post.

Does this make sense? Am I making an error? Am I overthinking this?

Thanks, for the help.