Deluxe Paint was an obvious extension of MacPaint for color work, but the ultimate version of it (written by the same people) was Studio/32 for the Mac (which also predates MSPaint). Let's not forget Photoshop and Fractal Painter predate MSPaint.
Deluxe Paint is not an extension/clone of MacPaint unlike most of the early paint applications of the era.
Dan Silva (the creator of Deluxe Paint) worked at Xerox before he got to EA. While there designed an in house paint application for the Xerox Star based on inspiration from SuperPaint by Richard Shoup (at Xerox Parc). When Silva joined EA in 1983, he wrote a version of Doodle for MS DOS for in-house use at EA - this port was named Prism. Deluxe Paint started as an Amiga port of Prism.
(Source: The Future Was Here: The Commodore Amiga, Jimmy Maher, The MIT Ppess Platform Studies series)
So they have shared heritage, given that MacPaint too was largely inspired by work at Xerox, and it's possible Silva made adjustments to Prism/DPaint after MacPaint was released since its release predated the commercial release of DPaint, but the application was already in use before MacPaint was released, though only in-house at EA.
If you look at the SuperPaint UI you can see its influence on some other painting applications too - Koala Painter on the C64 for example has a separate tools page that looks very close to SuperPaint.
But they are fundamentally different in other ways too - DPaint was designed specifically as a tool for artists first, and turned into a product afterwards. You see the difference in the various painting tools and brush support etc. that means you can actually "paint" with DPaint in a way you most certainly can't with MacPaint. The workflow is very different - you can do great things with MacPaint (or MS Paint) too, but it is far more laborious because it's hard to do much freehand drawing with it.
For me, when I moved from the C64, where I'd used Koala Painter - similar in capabilities to MacPaint - to the Amiga and started using Deluxe Paint, I first started drawing the same way I'd done on the C64: Laboriously placing pixel by pixel, with the occasional line draw or flood fill. Freehand was easier than on the C64 thanks to a mouse instead of a joystick, but the tools were not a good fit for that other than for large surfaces. But then I started experimenting with DPaints tools, like smear and blend, and my drawing style changed and got much more fluid and relaxed and it actually carried over to paper as well.
The history is interesting, but the influence of MacPaint on DeluxePaint was absolutely plain. (DeluxePaint clearly had some innovations over MacPaint.) Again, Studio/32 (and Studio/8 and Studio/1) were the full-circle -- the best of everything. Indeed, I often wish Studio/32 had had greater influence than Photoshop, which had and has a far inferior UI.
Source: I used all of these programs at the time, and spoke with people at EA who were familiar with the development of Deluxe Paint and Studio/x.
How exactly? Pretty much all of the functionality in MacPaint is in SuperPaint, which predated it by 6 years (I'm assuming you're not confusing them, but this is the Xerox SuperPaint, not the later Mac application of the same name).
There are some superficial UI similarities between MacPaint and DPaint in how they both switched from a big separate panel of tools to toolbars along the side, but that's pretty much the only similarities I can see between the two that were not already present in SuperPaint, and it's line with a general trend, and even there it's not all that obvious where DPaint draws most of its inspiration.
They differed substantially in that DPaint is geared towards making the entire screen available as a canvas, and most people I know who used DPaint spent most of the time with the UI hidden, only turning it on briefly to pick colours etc., or when using the split screen zoom (to date I hate the way most paint apps do zoom with a vengeance - DPaint had it right).
In many ways the UI of SuperPaint is closer to the UI of DPaint than MacPaint is. The windowed interface with multiple toolbars of MacPaint was totally foreign to Amiga paint apps that for years followed the Deluxe Paint model of putting tools firmly at the screen edges and making them easy to hide.
I don't see much influence from MacPaint that matters frankly, though I'm sure you're right there was some - MacPaint did after all make it out the door sooner and it'd be silly of them not to look at what was well received. But in the overall design, the influence from SuperPaint is plain - all the functionality shared between MacPaint and DPaint was there in Xerox' SuperPaint years before.
Other than their shared heritage with SuperPaint and superficial UI stuff, to me the two are fundamentally different - the "some innovations" is what makes DPaint interesting at all: MacPaint is not usable as a paint application; DPaint is. By the time DPaint came out, the market was flooded with MacPaint clones. But pretty much all of them lacked the paint tools that made DPaint exciting.
I'm not sure where Studio/x comes in - they post-date DPaint by several years as far as I can tell (wow - that's a hard product line to Google; I'd never heard of them before); the field was extremely crowded by then.
