Guys let me hopefully shed some light on this.
I worked for FedEx for 3 years and while I know UPS is a completely different company, they offer comparable express and ground services and I know their shipping POS to be similar (well, at least they used to be years ago.)
When any customer comes in on a Friday wanting to ship something "overnight," you as the shipping agent are presented with options.
"Overnight service"
"Overnight service with Saturday Delivery"
The DEFAULT service is just "Overnight" which returns a commit date of the following Monday (unless it's a holiday.)
If you want overnight service with Saturday delivery, you specifically have to CHOOSE that option. This is because there is a surcharge for Saturday delivery. (I remember at FedEx it was $12.50 for an overnight envelope above and beyond whatever the freight charge was.)
These options will either be available or not available in Sprint's "system" depending on what they have setup with UPS/FedEx for shipping options. If Sprint doesn't want to ship overnight with Saturday delivery as a matter of policy, that option will not even be available in their "system."
When you call UPS and ask them if they have Saturday delivery in your city, and they say "yes", that is accurate. But as mentioned above, it is up to the shipper to select that option and pay the surcharge, as applicable.
So if you are confused as to why UPS is telling you one thing about Saturday delivery and Sprint is telling you another, that is why.
Hope I haven't missed the mark on where the confusion is and that this helps to clarify.
EDIT: As far as "Saturday Delivery" stickers... let me address this also... Those are supposed to be applied to Saturday delivery shipments to make them easier to spot for sorting/delivery personnel. What really dictates the method of service is the barcoded label. In my time at FedEx I never heard of anyone not getting their Saturday delivery because someone forgot to put a "Saturday" sticker on it. We're a bit more sophisticated than that people... lol