NMEA is NMEA, NOT RS anything
Quoting from the NMEA 0183 version 2.20 standard:
"The drive circuit used to provide the signal 'A' and the return 'B' shall meet, at a minimum, the requirements of EIA-422-A (December 1978)". So in fact NMEA 0183 (from version 2 onwards) follows the RS-422 standard specifically.
The confusion comes because version 1 was based on RS-232 but with a wider voltage range. RS-232 uses ground as the return path while RS-422 uses a dedicated return path typically labelled 'B'.
This makes connection of devices following the different versions more difficult. It is further complicated with some devices having the required opto-isolated inputs and others not.
Another problem is most manufacturers don't bother informing the user which version they are following and even cherry pick the cheapest option from each version. Just because it is brand new does NOT mean it follows the latest version of the standard.
Yes, you can convert, but you can't just plug an NMEA output into an RS input or vise-versa no matter how you jiggle the wires.
I do this almost every day without the need to jiggle anything, I just plug them straight in. Remember that a 'B' of a talker should NOT be connected to the ground of a listener.
Modern RS232 chips work beyond the spec and can cope with RS422 signals. I have never yet found an RS232 connection not to work with an RS422 (NMEA 0183) input but it is not guaranteed. Also, due to the change of spec between version 1 and version 2 the latter versions are required to cope with voltages above the RS422 so they can handle direct (un-converted) RS-232 inputs as well. The only thing to watch out for is ground loops.
The information in NMEA 0183 is basically just text so if you are only interested in the data then it doesn't matter which RS standard you follow to get the text from one place to another (for instance Airmar smart transducers use RS-485 to transmit NMEA 0183 sentences).
If you are not sure about you connections ask in the forum and someone will OK it for you.
With both devices using A and B signals that is not your problem, both are following later than version 1, for the voltage levels at least.
going (from 0) pulses, 2V ..... I took the measurements at the input plug of the h3000
It sounds like you took the measurement across the A and B not across ground and A, can you confirm this? How did you measure the voltage? Using a meter on an active signal line is not always accurate.
When you had the h1000 connected was that using A and B for inputs or was it A and ground? Connecting a driving B signal into ground could have worn out the driver circuit of the Furuno. You may be able to rescue the situation using an isolating buffer box that can handle the low 2v input signal boosting it for the H3000 to read.