What would you want more?
A toy with one mechanic in an environment with many interactive mechanics?
A toy with many mechanics in an environment with very few interactive mechanics?
What kind of toy would you want?
I'd say the first option.
Bowling is a good example, a bowling ball is a very simple device, but a lot of environmental factors influence it. Wax on the floor, if the floor is level, the arrangement of the pins, and the strength of the person using it.
The second option is also interesting.
Computers for example. Very few environmental factors affect how the computer works, yet it is very complex in its own workings.
Small number of simple mechanics used in many different ways.
Probably relevant: Extra Credits discussing depth vs complexity
In addition to that video, it is something many different design sites and design blogs cover in depth.
Portal is an amazing game from a design perspective. if you watched the Extra Credits video link above, you can see it has a lot of depth with an extremely narrow mechanic.
In fact, it has ONE major mechanic: You can connect two points in space.
Both the first and second games were entirely based on experiments on that single mechanic. Reconnect where things fall. Reconnect where things slide. Reconnect where bullets go.
The OP's second option, having lots of tiny mechanics, is less good for design. While it does add combinatorial depth, the tradeoff is fairly shallow. When you have n choices it gives you 2^n combinations. Or, if order of your combinations matter, it gives !n combinations.
Say you've got ten options, that means the player must understand and choose between either a thousand different interactions if order doesn't matter, or choose between over three million different interactions if order does matter. Complexity grows incredibly fast, the depth gained probably grows far more slowly.
Don't see why this was moved from Game Design; it's a totally valid design question.
I think it depends on what you're going for, what you want to call attention to. If the goal is showcasing an awesome environment, putting the mechanics in the environment makes sense. Keep the player's eyes moving and exploring the environment to see what neat stuff there might be.
That might not be the goal, though; the goal might be "look at this thing, look at what it can do, don't you wish you were as awesome as this?" Then it makes sense for the abilities to be in the toy itself. Say it was Batman. While Gotham is a lovely place, it's not about Gotham and what neat things there are in Gotham, where Batman's just some schmo taking advantage of the environment. He's the goddamn Batman.
It depends what the fantasy is: "being somewhere" or "being something".
An example I like is bomberman. The simplest thing ever, placing timed bombs, and it allows for so much strategy and gameplay fun.
In many games I've truly enjoyed there is simple item with one or two mechanics that forces player to get creative and act with the environment. When learning to use something complex is patience put to the test and it may get quite confusing which is often too much for some to stay interested.
Depending on how you define the "environment" and the "toy", an example of the "toy with many mechanics in a sparsely-interactive environment" might be the traditional RPG: the player character is the "toy", and has access to an array of (potential) mechanics via the various skills and spells available, while objects in the environment tend to fall into a limited set of categories: merchants, NPCs, enemies, loot and "buttons" (that is, things to which a universal "use" command applies: doors, levers, etc.).
(One could define the field of skills and spells as an "environment" in which the player acts, but I suspect that my above interpretation is likely closer to the original intention.)
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