This webpage claims that people were scanning for #heartbleed before the April 7 disclosure of the bug. I doubt it. It was probably just masscan.
The trick with masscan, which makes it different from all other scanners, is that it has a custom TCP/IP stack and a custom SSL stack. When it scans a machine for SSL, it abruptly terminates the connection in the middle of the handshake. This causes no problems, but it's unusual, so servers log it.
The #heartbleed scanners have even less SSL. They just dump raw bytes on the wire and pattern-match the result. They, too, abruptly terminate the connection in the middle of handshaking, causing the same server log messages.
Masscan is really good at doing SSL surveys of the entire Internet, and a lot of people have been using it for that. The tool is only 6 months old, so it's something new in your logs. Two new things producing the same error messages might seem like the two are correlated, but of course, they aren't.