No, improvised weapons and weapons are different things
An improvised weapon is any object you could reasonably hit someone with, such as a candelabra, a crystal ball or the leg of a table. If we look at the full quote you are referring to:
In many cases, an improvised weapon is similar
to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option. a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
This implies that the table leg is still an improvised weapon, but you can use your proficiency bonus with it if you are proficient with clubs. It does not mean that the improvised weapon is now a ‘normal’ weapon.
What this means is that, because a table leg is not a weapon, it is an improvised weapon, the rune knight can not inscribe runes on it.
Whilst no longer offical, Jeremy Crawford’s tweet about eldritch knights and improvised weapons provides some support to this idea:
Weapon Bond works with a bona fide weapon ("Behold, my sword!"), not an improvised weapon ("Look, a stool!")
Whilst not perfect, as it doesnt use an example that could possibly be a weapon, it does give some insight into what the designers might have intended - that being features that require weapons to work need actual “bona fide” weapons, not improvised weapons.
Also, some colloquial support, Tavern Brawler grants you proficiency with improvised weapons. If the improvised table leg became an actual weapon, rather than being an improvised weapon, you could not apply your proficiency with improvised weapons to it as its “too weapon like”.
This makes very little sense: you can apply your proficiency bonus to a stool, but as soon as you break a leg off that stool, you can’t anymore because now its too much like a club?
What makes more sense is if: the broken chair leg remains an improvised weapon but, because its similar enough to a club, you can apply your proficiency bonus with clubs if you don’t have proficiency with improvised weapons.