I like these ideas, having a form in common would be a good way to tie the different types together while having the unique forms to each path. I hadn't even thought of
primal savagery, most of my thoughts on a damage spell was the warden using
shillelagh but
primal savagery certainly fits well. I would probably give the common form resistance to bludgeoning, slashing, and piercing damage as many of the forms in 4e granted some level of DR so resistance and primal savagery might work well as the basic warden form.
I think we could probably still keep the 6th level ability to cast
tidal wave while in a spellcasting form.
I'm trying to remember what I meant by enhanced forms

I think it would be something like:
Storm Warden
Level 2: Resistance to Thunder/Lightning. Deal additional thunder damage when using the attack action.
Level 10: Gain a flight speed equal to your walking speed while in your primal form.
This way, the Warden will essentially only have 2 forms throughout their career, the common tanky damage form and their specialised form which becomes more powerful as they level up. What do you think of this?
Yes to this.
Limiting the Warden to just the two forms, one generic and one specialized as you've delineated here, works well and gives us a solid structure to build on.
Currently the regeneration ability at 14th level is only active while in a primal form, however, it might be better as a specific primal form available to a warden instead of a general feature. For my basic warden V1.1 I've moved it to a primal form and given them Primal Convergence at level 14 which lets the warden spend two uses to activate two forms at once. I should probably post this revision as well. Primal Convergence only really works if forms don't really get more powerful as they level since if they do then Primal Convergence could become quite overpowered.
For our specialized warden, I'm not sure if Primal Convergence would be a good ability to have since they should have more limited numbers of forms to choose from.
Agreed. Let's strike the idea of being able to combine the features of two different forms at the same time. Having the level-specific enhancements for each primal form is more balanced and flavorful, and avoids building in unwanted complexities in play.
I hadn't considered this, I think we can include an extra ability at 18th level which replaces beast spells, at the very least, perhaps we could grant them 'Primal Focus' which allows the warden to ignore the need for material components. At 18th level though, that is a fairly minor ability to be gaining. Will have to think more on this.
"Primal Focus." Oh, boy! I've been unable to come up with an idea for replacing the Beast Spells feature, but this satisfies the parameters of utility and not being over-powered. I salute your little grey cells, my friend!
I should also note that I had completely misremembered the druid's wildshape ability regarding the number of uses. They only get 2/short rest so I have edited this in the opening post. The main differences are that wildshape uses an action, primal form uses a bonus action in a fashion similar to a bladesinger.
The reason I'd proposed increasing the uses of Primal Forms in an earlier comment is because of one of the other descriptors in Wild Shape. The paragraph under the Beast Shapes tables states, "You can stay in a beast shape for a number of hours equal to half your druid level (rounded down). You then return to your normal form unless you expend another use of this feature."
You limited the duration of a primal form to 1 minute, sufficient to most combat encounters. THIS IS A GOOD THING. Why?
I view primal forces as being very powerful, with the potential to rend a mortal mind apart. Keeping it under control is not easy. Limiting its duration represents that in simple way.
If we think of the evolution of forces and life in the game's multiverse in terms of increasing complexity and diversity, it might be something like the individual elements (most basic) > primal (first intermixing of life and interacting elements) > material world (the teeming complex life forms as we commonly know them, scattered planets, stars, galaxies, and the spaces between them, etc.). Where there's the Elemental Plane and the Material Plane, there is no Primal Plane (I haven't looked to see how that fits into the game's cosmology, but bear with me here): primal forces develop in and from the flux between the chaos of the Elemental Plane and the more limiting structures of the Material Plane.
So, how do we balance a primal form that lasts for only a minute versus a beast shape that can last for hours?
Well, one thing is that we could give primal form the same duration as wild shape. For a primal form that can take on a tree-form, that even makes a certain amount of sense. Having only 2 uses is plenty when each use gives you many possible opportunities to use your beast shape abilities over the course of a game day.
The enhancements we're looking at for the primal forms, though, seem to me to be different enough that keeping the duration to a minute, but increasing the number of uses as the druid goes up in level, is a reasonable option.
More food for thought.