I do believe the second statement refers to payload (The shortest possible TCP/IP packet is 40 bytes long - 20 bytes for TCP header + 20 bytes for IPv4 header).
Regarding the first, here's a quote from RFC 2663:
End of session for TCP, UDP and others
The end of a TCP session is detected when FIN is acknowledged by
both halves of the session or when either half receives a segment with
the RST bit in TCP flags field. However, because it is impossible for
a NAT device to know whether the packets it sees will actually be
delivered to the destination [...] the NAT device cannot safely assume
that the segments containing FINs or SYNs will be the last packets of
the session [...] Consequently, a session can be assumed to have been
terminated only after a period of 4 minutes subsequent to this
detection. The need for this extended wait period is described in RFC
793 [Ref 7], which suggests a TIME-WAIT duration of 2 * MSL (Maximum
Segment Lifetime) or 4 minutes.
Reference: https://www.rfc-editor.org/rfc/rfc2663
To my understanding, any packets that identifies a session would reset the TTL counter - but that depends heavily on implementation, since 'data' can be understood as 'packet' (minimum 40 bytes) or 'packet payload'. Nonetheless, @CodeCaster is spot-on; never assume that a connection is alive, make sure it is before sending (and, if possible and depending on criticality, acknowledge receipt.)