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I am working on a game where the player is one of very very few people who own spaceships capable of jumping between planetary systems. Different planets have vastly different technology, but roughly 50% of them are just slightly beyond the technology level we possess in real life right now. Others only possess older technology level anywhere from "modern" to stone-age.

I need a term for planets with our current technology level without implying it's the most up-to-date. I could just name them something abstract, but I want players to understand what the term means without having to be told.

What is a term I could use here that doesn't imply the planet's technology is up-to-date and that players would understand intuitively?

This is a good chance for you to invent a future present adjective for a society which will inevitably exist on the futuristic socio-economic scale, I predict. Done cleverly, without any doubt the descriptive term will be plagiarized. I’d advise to think in terms of future “vintage” or future “primitive” - or, i.e., what design now seems extremely technologically advanced to even the most classified of governmental technology project managers will, at some point in the future, have lost its usefulness, but not its ability to perform the original task. To the classified military project manager of the future it is nothing but scrap. But to a future society with few resources, it can be a life sustaining blessing. In what terms would you describe such a society ??

There may be several ways to go here:

Post-nuclear but pre-fusion.

Subsequent to the development or use of nuclear weaponry; specifically
of or belonging to the period after a nuclear war.

pre-fusion needs to be broken down to pre- prefix:

a prefix occurring originally in loanwords from Latin, where it meant
“before” (preclude; prevent); applied freely as a prefix, with the
meanings “prior to,” “in advance of,” “early,” “beforehand,” “before,”
“in front of,” and with other figurative meanings (preschool; prewar;
prepay; preoral; prefrontal).

and fusion - suffix:

Also called nuclear fusion. Physics. a thermonuclear reaction in which
nuclei of light atoms join to form nuclei of heavier atoms, as the
combination of deuterium atoms to form helium atoms.Compare
fission(def 2).

Since there is no conditional probability that your world will enter a "post-truth" era, I'll dismiss this possibility.

Parochial space-faring society:

very limited or narrow in scope or outlook; provincial

I'll introduce a further possibility.

A "nearly planet-bound" society or culture.

The meaning being explicit with regards the culture or species being almost entirely on one planet, the implication being that no colonies have been established elsewhere in great number.

I like "the silicon age" commented but different planets might develop the same abilities by using different discoveries / inventions / technologies. For example the computer was invented before the age of silicon, which was the facilitator. So is it the technology or the ability which is important?

I think the latter, so I propose

The age of spaceflight.

or

Planet X has reached the space age.

which says what can be done but not how.

You might look at what other games have done. Civilization VI breaks human history into these eras:

Ancient Era (4000 BC)
Classical Era (1600 BC)
Medieval Era (120 ~ 200 AD)
Renaissance Era (1100 ~ 1200)
Industrial Era (1625 ~ 1675)
Modern Era (1840 ~ 1860)
Atomic Era (1920 ~ 1950)
Information Era (1960 ~ 2000)
Future Era GS-Only (2020 ~ 2050)

(I believe the Civilization Wiki got the dates wrong, but the names are reasonable.)