...but if the players search for traps, how is that resolved?
I have long held that the simple "search" check is the most complex thing in all of D&D and this doesn't really matter whether you are talking about "I check for traps' in 1e AD&D or "I search the 5x5 grid" in 3e D&D.
The problem is we are trying to take something that is concrete and abstract it. The character did a series of progressively more intrusive things in some area that is not really specified by the mechanical declaration by the player to avail themselves of the rules. This puts an enormous burden of interpretation on the GM. Did they look or did they touch? What things did they look at or touch? In general, I tend to require the declaring character to make some sort of declaration of intent and method when "searching" in order to answer the questions you are answering. For example, a visual inspection only tends to give a penalty on the search check while being generally safer than tactile inspection.
In general, a search for traps is intended primarily as a search for a trigger mechanism, and not the trap itself. In the case of the two being close together, the searcher may find both, but if the trigger mechanism triggers something remotely they tend to find the trigger mechanism but not an explanation of what it does. In the case of the pattern of small holes in the wall, a search of the wall my find that relatively easily (DC under 20) but barring ability to gather more information (like microscopic vision or x-ray vision) what is down them small holes might not be obvious. That it's a trap could be inferred but wouldn't be something I'd tell them. They are gathering information they can find.
Finding the trigger mechanism however might be more difficult (DC over 20) and might require searching a different area. This might discover a pressure plate or an additional lever in the chest latch or the lock mechanism, or whatever. This is a "trap" in the sense the Rogue recognizes this as a trigger mechanism but doesn't tell the rogue what it does necessarily unless the trigger is closely related to the trap.
Let's say those questions are resolved, how do the characters handle it? Let's say they (somehow) know for certain it's a trap. Do they make a skill check to disable/bypass it. Do they have to describe how they disable/bypass it? If the latter, do they also have to make a skill check, or does the GM adjudicate that, possibly granting automatic success or failure?
If disabling device, they don't have to describe how they disable the trigger. Note however in many cases they can describe how they evade the trap without disabling the trigger. For example, having found a pressure plate, they might describe how they outline it in chalk and then say, "No one step here, there is some sort of pressure plate.", in which case in normal circumstances everyone will carefully step around the trap and not set it off. However, forced movement from panic, bullrushes, or moving into that square while in combat might result in a percentage chance of accidentally setting it off.
A lot of the time if the player can identify the trap and the trigger, the trap can be "disarmed" by having it go off in a controlled manner. This is what Indiana is basically doing when he first finds the pressure plate for the dart trap. This doesn't require a disable device roll. If the PC's can identify a trip wire and observe from the walls it's probably some sort of scythe trap, they might cut or pull the trip wire with a 10' pole or a polearm from a relatively safe distance and hope for the best. If they suspect a door is trapped, they can always try to open it in a way that evades the trap, such as tying a rope to the handle or opening it from the side. Of course, the trap might always turn out to be different than what they expected if they can't identify what it is only the mechanism.