Is there a way to discern exhaustion due to altitude sickness from exhaustion due to the climb itself? Of course they go hand in hand but the lower oxygen levels would make you feel even more exhausted than near sea level. I climbed a mountain almost 7,000 ft (2.1 km) high this year and it was a steep climb near the summit, I felt very exhausted and had to make breaks very often and sit down thereby. However, I can't tell if the less oxygen at that elevation enhanced it. Sometimes I was hyperventilating, but again I dunno whether it was due to the steep climb only or due to the high altitude as well. When at rest, expecially after reaching the summit, things came back to usual eventually.
The single most obvious indicator is going to be looking at your SpO2 level. If you oxygen saturation is low it is a good indication you are experiencing altitude sickness. The easiest way is with a pulse oximeter. They cost <$20 on Amazon and are both small and light. If you are going to be doing something outdoors where altitude is a concern then just bring one with you. I bring one with me anytime I am sleeping above 9000 feet since I'm at higher risk having gotten HAPE before.
The general signs/symptoms of AMS (acute mountain sickness, the technical term for mild altitude sickness) are going to be headache, nausea, fatigue, loss of appetite, and trouble sleeping (less useful during a hike). You will notice none of those symptoms are particularly unique to AMS. As such it is very difficult to conclusively say if a patient has AMS. You are often going to look at context clues (altitude, rate of change of altitude, sleeping altitude, temperature, hydration history, etc) to make an educated guess if it is AMS or something else. Without a pulse oximeter or other advance equipment a definitive diagnosis isn't possible.
When to be more concerned? If AMS progresses to HACE (cerebral edema) or HAPE (pulmonary edema). HACE and HAPE are both potentially life threatening and are cured by immediate rapid descent but can be temporarily mitigated with supplemental oxygen. HACE is typically indicated by ataxia and altered mental status. HAPE is typically indicated by severe shortness of breath, severe fatigue, and a "wet" cough.
As with all medical topics, I want to stress that you are best served by taking a WFA/WFR course. Real training cannot be replaced by simply reading online forums.
An easy way to tell is simply that your elevation was not high enough to cause even mild altitude sickness. The cabin of a passenger jet has an air pressure equivalent to about 8000 ft of altitude, so there's no way that 7000 ft will cause any kind of altitude sickness. Elevations more like 11,000-13,000 ft are required to produce any significant problems for most people, and life-threatening or totally disabling problems are only likely at even higher elevations.
Blinded studies have been done with people who didn't know what elevation they were at, and it was extremely common for people at essentially sea level to report problems that could have been diagnosed as AMS. This is because the common symptoms of mild illness are nonspecific ones like headaches and fatigue, which can just result from sleep deprivation, unaccustomed exertion, or sudden caffeine withdrawal.
It sounds a lot like exertion, and the reduction in partial pressure of oxygen could easily have contributed, without meaning you were ill.
At this sort of altitude (7000ft/2000-2500m) you're getting about 4/5 the sea-level amount of oxygen per breath. Very very roughly, you might consider that you have to exert yourself 20% less hard to get out of breath - that's quite noticeable. This caught me out after travelling pretty quickly to Yellowstone from near enough sea level, when I ran up a flight of steps and ended up gasping for a minute or two. I've also had a similar experience to you cycling in the Alps, rising in a few hours from ~900m to 2400m.
That shortness of breath alone isn't enough to call it AMS. A headache in combination is starting to get a bit more worrying, but dehydration can easily cause that, and it's easy to get dehydrated when ascending a lot under you own power, whatever the actual means.
The cautious approach is clearly to fear AMS* and go down, moving steadily and safely.
However at these marginal altitudes, rest (+fluids, even some food) is a reasonable approach, especially if you're fit. This is because in fit individuals the after-effects of exertion will soon pass, while AMS doesn't respect fitness levels. So for example your pulse rate should come down pretty quickly if there's nothing wrong with you and any lightheadedness should soon pass.
Then you can come to a decision, which will also be influenced by the routes ahead and behind. For example, continuing onwards and upwards is far more reasonable if it's followed by a proper descent than if the climb puts you between two passes, with the only way out to go over one - that's bad if it did turn out to be AMS after all.
Note that if serious symptoms of HACE* or HAPE* start to appear you should descend immediately and seek medical help. These would include:
- shortness of breath even when resting,
- a headache that doesn't respond to treatment (so if you have a headache and no other symptoms, it might be reasonable to take what you'd normally take for headache, rest for long enough that your painkillers should take effect, then decide)
- confusion and unsteadiness. The affected person may not realise they're not well, so look out for your companions (in this respect hypothermia is similar)
* AMS: Acute mountain sickness. HACE: High altitude cerebral oedema. HAPE: High altitude pulmonary oedema (note that the abbreviations derived from American spelling are used even in British English)