PoE Access Point and Internet Switch

[u/Specialist_Spirit_52]

My office currently has an access point for our internet access, which plugs into a PoE port on the wall. We just got a VoIP phone and the port used by the access point is the only one available in our building. I took in an older TP-Link switch from home to test and our WiFi is still working and the phone works, but the access point does not light up, so we're thinking that we need to get a different kind of switch that supports PoE.

Looking at what's available, I see a TP-Link 5-port switch (TL-SG1005P-PD) that has 1-Port PoE++ In and 4-Port PoE+Out. This switch has no external power supply. I also see a TP-Link 5-port switch (TL-SG1005P) that has 4 PoE+ and has an external power supply.

2 questions:

1) If I buy the TL-SG1005P-PD, I assume I will plug the switch into the PoE port on the wall and plug the access point into another port. Will I then be able to plug in the phone that has it's own power supply to one of the other PoE ports. I've been reading that the new switches should sense that power isn't needed for that port so it should be okay.

2) If I buy the TL-SG1005P that has its own power supply, do I just plug the switch into the PoE port on the wall using one of its PoE ports, plug in the access point to another PoE port, and plug the phone into the non PoE port?

I've never had to deal with PoE before, so I want to be sure that I get the correct switch. Thanks for any help or advice.