A bug in my high school physics intuition

I think of myself as having pretty good physical intuition, at least for plain old mechanics. Some systems are hard to get a grip on (e.g. why does a hard-boiled egg stand up when you spin it?), but at least I recognize when I'm confused about them. However, I recently got some simple mechanics really wrong, and didn't feel the slightest bit of confusion about it. So I went on a quest to update my intuition so that the same failure mode never happens again.

Here's the motivating question.

You're sitting on a bicycle or in a rocket ship, stationary. This gets boring after a while, so you accelerate to 1 m/s. After cruising for a while you get bored again, so you accelerate some more, up to 2 m/s. What took more energy, getting from 0 m/s to 1 m/s, or getting from 1 m/s to 2 m/s?

The problem is that there are two intuitively compelling answers, which severely conflict.

Answer 1. Once you're cruising at 1 m/s, we may as well use the inertial reference frame which is moving at 1 m/s, in which you are stationary. Then the second spurt of acceleration corresponds to speeding up from 0 to 1 m/s, which is exactly what the first spurt did in the stationary reference frame. So the two take the same amount of energy.

Answer 2. Just measure the kinetic energy differences. Kinetic energy is proportional to the square of speed, so the first spurt of acceleration took $1^2 - 0^2 = 1$ unit of energy and the second one took $2^2 - 1^1 = 3$ units of energy, which is 3 times as much energy.