Spam and Unsolicited Bulk Email (UBE): Linode.com has a zero tolerance policy on spam, Junk E-mail or UBE. Spam, Junk-mail and UBE are defined as: the sending of the same, or substantially similar, unsolicited electronic mail messages, whether commercial or not, to more than one recipient. A message is considered unsolicited if it is posted in violation of a newsgroup charter or if it is sent to a recipient who has not requested the message. UBE also includes e-mail with forged headers, compromised mail server relays, and false contact information. This prohibition extends to the sending of unsolicited mass mailings from another service, which in any way implicates the use of Linode.com whether or not the message actually originated from our network.

Mailing Lists: Linode.com's mass mailing rules also apply to mailing lists, list servs, or mailing services you may contract with. The policy is stated as follows: An acceptable mailing list will be focused at a targeted audience that has voluntarily signed up for your e-mail information or that has made their e-mail address available for distribution of information from you. The list must also allow for automatic removal of recipients with non-distribution in the future.

Spam and Unsolicited Bulk Email (UBE): Linode.com has a zero tolerance policy on spam, Junk E-mail or UBE.

Mailing Lists: Linode.com's mass mailing rules also apply to mailing lists, list servs, or mailing services you may contract with.

If you feel the need to ask the question, you probably need to think carefully about the answer.

What about systems that are used by people that sign up and or pay to be notified by email and/or SMS via email? Those can result in relatively large volumes of email traffic, would that be considered SPAM?

Probably not (though IANAL, and clearly there is quite a continuum of possibilities within the above), assuming that it fitted well within the Linode ToS.
I suspect that you really would need to know what you're doing though, and be thoroughly familiar with the steps needed to ensure that the system wasn't misused (by bad users, random bots etc.).

You may also be interested in the Spamhaus definition of spam: http://www.spamhaus.org/definition.html.

The fact of the matter is that they lost their trademark. They can say what they want, but legally they can't do anything about people using the term SPAM whatever way they want.