The word "matching" can sometimes be vague. Stuart (2010) defines matching "broadly to be any method that aims to equate (or "balance") the distribution of covariates in the treated and control groups". This would include PS matching, PS stratification, and PS weighting, but not PS as a covariate. Ho, Imai, King, and Stuart (2007) consider matching as nonparametric preprocessing to reduce model dependence. This would include PS matching and PS stratification, but not PS weighting or PS as a covariate. More specific definitions of matching refer to matching as subsetting (and optionally, but typically, pairing) the treated and control groups in order to equate the distribution of covariates in the treated and control groups, in which case only PS matching fits this definition. Don't get too hung up on the definition of "matching"; just try to be consistent and clear when describing what you are doing.
The four methods you mentioned all involve the propensity score, but that doesn't make them all matching methods. PS as a covariate is essentially a regression method just like covariate adjsustment but summarizing all the covariates into a single value. This method is almost always used improperly and has poor statistical performance when used improperly, so I recommend you do not use it if you are just starting out.
PS matching methods offer flexibility with the number of units retained. It's also important to know that the number of units retained is not the sole measure of the precision of the sample; for example, full matching retains all units but that doesn't mean no precision is lost when using it. Different ways of performing the matching can yield different numbers of remaining units. The simplest case is 1:1 nearest neighbor matching without replacement and without a caliper. This involves matching one control unit to each treated unit, so the number of remaining units should be twice the original size of the treated group. You can also add a caliper; this involves disallowing any pairs that are farther apart than some threshold set by the user. Any units without a close enough match will be discarded, so using a caliper will result in fewer units remaining. Matching with replacement allows each control unit to be matched to multiple treated units. If one control unit is a good match for many treated units, then fewer control units need to be used to find matches for all treated units, resulting in fewer units remaining. K:1 matching involves matching more than one control unit to each treated unit; this obviously increases the number of units remaining.
Different matching methods will produce different balance statistics. The whole point of matching is to try many matching methods and find the one which achieves the best balance in your dataset without discarding too many units. It is critical that you do not involve the outcome variable in this process.
The articles on the
MatchIt website attempt to provide a tutorial for the use of matching, as well as descriptions of many matching methods, best practices in balance assessment, and effect estimation after matching. They include many references, which you should read before proceeding.
One thing I want to stress is that propensity score methods are advanced statistical methods that require highly specialized training to use correctly. Every single systematic review of the use of propensity score methods in published research indicates that many researchers are using the methods incorrectly. Tutorials may help you understand the basics of using the methods in the most ideal cases, but the most basic uses of these methods often yield the worst performance. I highly recommend you consult or collaborate with a statistician trained in the use of these methods before proceeding on your own.