I'm worried about whether I qualify as being removed or deported
You were "removed", not "deported". To get very technical about it, you were issued 'administrative removal' under Paragraph 320 of the Immigration Rules because you failed to qualify as a visitor under Appendix V of the same rules.
and if there is an ensuing ban in place.
There is no ban in place. Were there a ban they would have served you different paperwork and you would be aware of it. They don't ban someone during a portside removal anyway (except in the most egregious cases like forgery, identity theft, contraband and the like).
Can you offer any insight? Does this count as deportation, or simply a matter of course during the refusal of leave to enter?
Assuming that your visit was to maintain a relationship with your 'sponsor', then the latter would apply. It's fairly normal for people pursuing long distance relationships to try and conceal this during their landing interview or to have discrepancies in their story. They do not consider that sort of thing to be the type of deception that attracts a ban, but Immigration Officers get very annoyed when they are being lied to.
You will need to report your removal when you apply for entry clearance.
Your removal notice has your 'Port Reference' number on the top, don't lose it. You will need to provide this when you apply (or optionally add several weeks to your waiting time).
Secondly, how much money is necessary to have evidence of for a one month visa?
There is no prescribed amount but it should properly agree with the purpose of your visit. If you are planning to cohabit with your 'sponsor' then you wouldn't need a lot for maintenance and accommodation.
Can you offer any insight?
You wrote that you are applying for entry clearance, this is recommended (but not required) following any sort of removal or refusal. The big problem I see in your narrative will be establishing that you are a genuine visitor. Most visitors plan to stay for about 2 or 3 weeks because they have jobs and other commitments to attend to, and part of your challenge will be to prove that you have those types of commitments. It's safe to assume that you do not because your landing interview would have turned out quite differently were you able to present a coherent picture of a bona-fide visitor to the Immigration Officer. Also, showing up in the UK without a return ticket is pretty much an open admission that you do not have strong ties to your home country, and that's something they will look for.
I am now attempting to make a visitor's visa to return to the UK
That is the recommended strategy for your situation. To apply you would create an account at Visa4UK and fill out the form. A UK Standard Visitor Visa (which is formally and technically called 'entry clearance') will reduce the problems of arriving in the UK and your landing interview will be more of a formality. The first page should be filled out like this screen shot...
For the purpose of your visit, you can explain that you are maintaining a long distance relationship and include a statement from your 'sponsor'. It is also emphatically recommended to wait before hitting 'submit' until your circumstances are stable and your homeland ties are plainly evident. If these are not in order, your application will be refused and then matters will be worse...
Any suggestions for what to include as far as evidence when I make my visa application?
At this point you can plan to spend some time carefully studying the guidance along with Appendix V of the Rules. The guidance will help identify what evidence they will be looking for.
Note: The question about 'deportation' versus 'removal' is here.
Note: Related question with redacted removal notice on Expats.
Note: A similar question (but not a duplicate) in the TSE archives is here.
Note: Per the comment by phoog (to whom thanks), having a proper visa (entry clearance) in your passport before you leave your home country is different from getting 'leave to enter' when you arrive in port. Both of these options are available for Americans, Canadians, Aussies, and so on. I will concede that the UK is in a totally different orbit with respect to visas and terminology. Sadly, if you are going to conduct an LDR, you should familiarize yourself with it.