I've been asked to estimate jobs in every conceivable way. I've seen estimates go wrong in many ways.
I can hit any deadline with a fuzzy enough scope of work.
So, presuming a scope of work with some meaningful acceptance tests in it I'll tell you the secret to generating the only meaningful time estimates I've ever encountered.
Imagine the work you've been asked to do is harder than anyone expected. Imagine that it's already been however much time anyone thought it would take and it's still not done. Now how much longer are you willing to waste time on it before you're sick of this and want to try doing anything else?
That process generates the most realistic time estimates I've ever seen. Has nothing at all to do with the poorly defined, not yet understood problem. It's all about how patient we are with getting the solution this way before we want to tear up the whole plan.
The only way I've ever improved on this was to prototype a solution before the job ever starts so I already know it will work. Some people work cookie cutter jobs that are the same thing over and over. That's pretty much the same as prototyping. Those kinds respond best to detailed specs. Mostly because those jobs are about not forgetting to do something.
For SWAG it's really about how people feel. The reality around the problem. Not the problem. The job ends when it's done or people get sick of waiting and try something else. We're really estimating our patience with the problem.
As for communicating a SWAG estimate you've already taken care of one of the biggest sources of misunderstandings. By expressing time durations as jobs that are small, medium, large, and very large you prevent the perception that you said it'd be done in exactly 50 days. Such numbers are always fuzzy. Customers shouldn't be led to think that they aren't.
However you communicate it, listen to your customer. Try to get a sense of how well you're being understood. What seems perfectly obvious to your might come as a complete surprise to them. This process has enough uncertainty in it without adding needless surprises.
Sometimes the SWAG estimate is expected to be at the high end of a range. This form of estimate can make it difficult to manage customer expectations, a 15 day effort is very different from a 50 day effort, and the customer can green light a SWAG under the optimistic assumption.
This is exactly why I explicitly tie a SWAG estimate to how people feel about the work. If the customer is obviously uncomfortable with the idea that a task could be a 50 day effort then don't give into the temptation to sell it as a 15 day effort without nailing down the scope of work to something that fits well within a 15 day effort. Never let how you feel about time get compressed without saying what will be lost by doing that.
If the customer approves of a SWAG estimate, we can feel obliged to cap our detailed estimate at the high range of the SWAG. If we move up a range then there are usually additional billing discussions which are painful, slow, and lacking guarantees for compensation.
Well you are obliged. In fact you are obliged to not wait to start the additional billing discussions. If after the first day on a 15 day job it becomes clear it's going to be a 40 day job you the report this then. Not after 15 days. If the customer says no the responsible thing to do isn't to go crazy trying to get done in 15 days. It's to be willing to walk away.
Personally, I like to give the customer the best understanding of progress as I can. When I'm working with a deadline I give daily updates about how likely we feel it is that we'll meet the deadline. When I'm working a checklist on estimated work I encourage evidence based scheduling. That is, to use velocity to take how closely we're matching our estimates and extrapolate our completion date. This can be recalculated after every milestone. It's good to see this recalculated often.
Doing that doesn't do a thing to make the work go faster but it gives those waiting on it the feeling that they understand what's going on. Managing those feelings is actually the most important thing if you want them to keep giving you work.