The commend says 'floating ice' the link you point at says 'ice on land'. The commender is 100% right but apparently the HN crowd has a lot of people that just love to downvote (which is why downvoting should be banned).

The link the op points says:

"First, as the oceans warm due to an increasing global temperature, seawater expands—taking up more space in the ocean basin and causing a rise in water level."

The ice melting is not the cause of seawater warming. Ice melt may indicate seawater warming, but the OP is correct that in itself it does not affect sea levels.

Actually, because sea ice has a different albedo than exposed sea water [0], sea ice melting is a significant cause of seawater warming (seawater warming is also, of course, a cause of sea ice melting; this is a classic positive feedback loop.)

[0] https://nsidc.org/cryosphere/seaice/processes/albedo.html

Yes, the sea water absorbing more sunlight due to lack of reflecting ice could cause it to warm significantly. But this is indirectly caused by sea ice melting, and a more speculative connection between ice melt and water levels rising. Furthermore, I'd like to see hard numbers, since most of sea ice is under water and does not reflect sunlight.

Stick a piece of ice in a cup of water, mark the water level before and after the melt, and tell me what happens.

Thermal expansion effect in cup of water is a little smaller than the thermal expansion effect in the oceans

Thermal expansion is a separate issue. Ice melt in and of itself does nothing to raise water level.