When can 叫 be used saying that someone is calling someone and when for asking a name? From The HSK acadamy site, it seems that 叫 can be used in two contexts.
First, it can be used to say someone called someone else:


妈妈和爸爸叫我


There is no translation given, but I think it means mum and dad call me.
In the second and third examples on this page,


我的名字叫李明

小姐，你叫什么名字？



I think 叫 is here used for giving or asking someones name. I would translate the second sentence as My name is 李明, and the third as Lady, what is your name? However, this is always used together with 名字, which means name. However, I've also seen simply


我叫 Wythagoras


for simply my name is Wythagoras (not my real name, by the way).
I am a bit confused by all the different usages of 叫, so the question is: In what context can 叫 be used for saying that someone is calling someone (as in the first sentence) and when for asking a name, when should 名字 be used with 叫 and when can it be left out?
Your intuition is correct. 叫 can be used in both ways (and even more). Here are the most frequent ones.

1) to call, to summon (using someone's voice). Verb is transitive:


  我爸爸妈妈叫我 
  Mom and dad call me (they are calling my name)


2) stating one's name. Verb + proper noun or definition:


  你叫什么名字？我叫张三！
  What's your name? My name is John Doe!


3) to yell, to shout, or more specific verbs when referring to animals, e.g. to bark. This is a colloquial usage. Verb is intransitive:


  我狗在叫！
  My dog is barking


4) giving orders, asking to do something. Colloquial usage. Accepts a verb phrase:


  我的驾照过期了，我要叫他开车！
  My driver's license has expired. I'll ask him to drive!



Specifically, how to decide between the usage 1 and 2. It is often made clear by the context of the whole sentence, as customary with the Chinese language.
As a rule of thumb, you'll have 叫 (to call) when it is followed by a regular object phrase, and you'll have 叫 (asking names) when it is followed by a proper noun or generically by a definition.