“Speed” is ambiguous:

• “Fast internet” can refer to bandwidth, e.g. being able to stream dozens of 4K movies from Netflix at a given time.
• “Fast internet” can refer to latency, e.g. getting a response within milliseconds.

One does not imply the other:

• I can sign Morse code with a laser pointer to a few buildings further, which would be very low latency (speed of light), but extremely low bandwidth too (a dozen bits per minute).

• I can load a truck with 4 TB hard drives and drive it across the country, which would be very high bandwidth (a dozen petabytes per hour), but atrocious latency (hours).