About DH-EKE and ECDHE_PSK (PSK versus Pasword) I am currently having a look at J-PAKE , so learning about Authenticated Key Exchange :
Reading 2.2.2 E.K.E from this paper https://people.kth.se/~maguire/Erik_Ehrlund_Literature_Study_report_v1.2.pdf
I understand the use of the password to authenticate a DH handshake.

The question is :

Is TLS DH ciphersuite based on PSK authenticated in the same way as the DH exchange authenticated with password ?
Is entropy a matter for PSK considering weak password ?
Assuming you're actually asking wether there is a difference for TLS in between a (high-entropy) pre-shared key (PSK) and a (low-entropy) pre-shared password, the short answer is:

No.

The long answer, however is a bit more complicated.
By itself the TLS-PSK ciphersuites don't differentiate between different degrees of entropy. However there are a few TLS-SRP ciphersuites, using the secure remote password protocol - a password-authenticated key-exchange (PAKE) protocol. They actually have some nice properties which only get interesting if you can assume the PSKs to be low entropy.


  Does entropy matter for TLS-PSK?


Yes it does.
If you're using the TLS_PSK_... ciphersuites you actually use the PSK as a basis for your session key meaning that this key is susceptible to off-line brute-force attacks. However it looks like you want to use TLS_ECDHE_PSK_... ciphersuites, which are much harder to attack. You can't derive the session key without breaking the ECDH key-exchange and knowing the PSK. So there's no trivial passive attack to obtain the transmitted contents after the handshake was done. However active attacks are very well possible, as you can install yourself as a man-in-the-middle (standard attack on DH) and try to impersonate the other client by trying out the different possible values for the PSK. This is infeasible if the PSK is high-entropy but very well possible if the PSK has low entropy.